About the complex nature of the wave function? 1.
Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.
2. 
A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;) 
3. 
One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u  = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v  = \frac{\hbar^2}{2m} \partial_x^2u - V u$$
(..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$).
 A: More physically than a lot of the other answers here (a lot of which amount to "the formalism of quantum mechanics has complex numbers, so quantum mechanics should have complex numbers), you can account for the complex nature of the wave function by writing it as $\Psi (x) = |\Psi (x)|e^{i \phi (x)}$, where $i\phi$ is a complex phase factor.  It turns out that this phase factor is not directly measurable, but has many measurable consequences, such as the double slit experiment and the Aharonov-Bohm effect.  
Why are complex numbers essential for explaining these things?  Because you need a representation that both doesn't induce nonphysical time and space dependencies in the magnitude of $|\Psi (x)|^{2}$ (like multiplying by real phases would), AND that DOES allow for interference effects like those cited above.  The most natural way of doing this is to multiply the wave amplitude by a complex phase.  
A: This question has been asked since Dirac 
In fact Dirac's answer is available for $ 100 from JSTOR in a paper by Dirac
from I think 1935 ?
A recent answer from James Wheeler - is that the zero-signature Killing metric of a new, real-valued, 8-dimensional gauging of the conformal group accounts for the complex character of quantum mechanics
Reference is  Why Quantum Mechanics is Complex , James T. Wheeler
ArXiv:hep-th9708088  
A: The question is a good one and has been asked also by Ehrenfest (1932): "Einige die Quantenmechanik betreffende Erkundigungsfragen". The answer was given by Pauli (1933): "Einige die Quantenmechanik betreffenden Erkundigungsfragen". Unfortunately I'm not aware of a english translation of these two publications. However one can find a slightly different form of the answer also in Pauli's book "General Principles of Quantum Mechanics" p.16. In that book Pauli writes

a single real function is not sufficient in order to construct from
  wavefunctions of the form (3.1) a non-negative probability function
  that is constant in time when integrated over the whole space.

I will try to summarize his arguments here:
A wave packet to describe a single particle (basically deBroglie's idea) can be written generally like
$$
u(x,t) = \int U(k) e^{i(kx-\omega t)} dk = \int U(k) e^{ikx} dk \, e^{-i\omega t}
$$
where $U(k)$ is the Fourier transform of $u(x,0)$. The complex conjugate of this wave packet is
$$
u^*(x,t) = \int U^*(k) e^{-i(kx-\omega t)} dk = \int U^*(k) e^{-ikx} dk \, e^{i\omega t}
$$
One can also define such wave packets in electrodynamics. But in quantum mechanics we have an additional condition, namely that the probability $P(x,t)$ to find a particle must always be positive and the total probability to find a single particle somewhere must be one, so
$$
P(x,t) \geq 0 \\
\int P(x,t)\, dx = 1
$$
Pauli argues that the simplest ansatz to construct such a function $P(x,t)$ from $u(x,t)$ is a definite quadratic form from the functions $u$ and $u^*$, that means
$$
P(x,t) = a u^2 + b {u^*}^2 + c u u^*
$$
Now from the form of $u(x,t)$ and $u^*(x,t)$ we see that
$$
u^2 \sim e^{-2i\omega t}\  \text{and}\  {u^*}^2 \sim e^{2i\omega t}
$$
and an integral over space over these two functions can never be time independent. So the constants $a$ and $b$ must be zero in the ansatz for $P(x,t)$. Only the product of a wave packet and it's complex conjugate will yield a time independent total probability:
$$
1 = \int P(x,t)\, dx = \int uu^*\, dx = \iiint U(k)U^*(k') e^{i(kx-k'x)} \, e^{-i\omega t} e^{i\omega t} dk dk' dx \\
= \iint U(k)U^*(k') \delta(k-k') dk dk' = \int \left|{U(k)}\right|^2 dk = \int P(k)\, dk
$$
Since the product $uu^* = Re[u]^2 + Im[u]^2$ it follows that - as Pauli said - in order to compute a meaningful probability from wave packets of the form $u(x,t)$ one needs the real and the imaginary part of $u(x,t)$ and the wave function in quantum mechanics must be complex.
A: This year-old question popped up unexpectedly when I signed in, and it's an interesting one. So I guess it's OK just to add an intuition-level "addendum answer" to the excellent and far more complete responses provided long ago.
Your kernel question seems to be this: "Why is the wave function complex?"
My intentionally informal answer is this:

Because by experimental observation, the quantum behavior of a particle far more closely resembles that of a rotating rope (e.g. a skip rope) than it does a rope that only moves up and down.

If each point in a rope marks out a circle as it moves, then a very natural and economical way to represent each point along the length of the rope is as a complex magnitude. You certainly don't have to do it that way, of course. In fact, using polar coordinates would probably be a bit more straightforward.
However, the nifty thing about complex numbers is that they provide a simple and computationally efficient way to represent just such a polar coordinate system. You can get into the gory details mathematical details of why, but suffice it to say that when early physicists started using complex numbers for just that purpose, their benefits continued even as the problems became far more complex. In quantum mechanics, their benefits became so overwhelming that complex numbers started being accepted pretty much as the "reality" of how to represent such mathematics.
That conceptual merging of complex quantities with actual physics can throw off your intuitions a bit. For example, if you look at moving skip rope there is no distinction between the "real" and "imaginary" axes in the actual rotations of each point in the rope. The same is true for quantum representations: It's the phase and amplitude that counts, with other distinctions between the axes of the phase plane being a result of how you use those phases within more complicated mathematical constructions.
So, if quantum wave functions behaved only like ropes moving up and down along a single axis, we'd use real functions to represent them. But they don't. Since they instead are more like those skip ropes, it's a lot easier to represent each point along the rope with two values, one "real" and one "imaginary" (and neither in real XYZ space) for its value.
Finally, why do I claim that a single quantum particle has a wave function that resembles that of a skip rope in motion? The classic example is the particle-in-a-box problem, where a single particle bounces back-and-forth between the two X axis ends of the box. Such a particle forms one, two, three, or more regions (or anti-nodes) in which the particle is more likely to be found.
If you borrow Y and Z (perpendicular to the length of the box) to represent the real and imaginary amplitudes of the particle wave function at each point along X, it's interesting to see what you get. It looks exactly like a skip-rope in action, one in which the regions where the electron is most likely to be found correspond one-for-one to the one, two, three, or more loops of the moving skip rope. (Fancy skip-ropers know all about higher numbers of loops.)
The analogy doesn't stop there. The volume enclosed by all the loops, normalized to 1, tells you exactly what the odds are on finding the electron along any one section along the box in the X axis. Tunneling is represented by the electron appearing on both sides of the unmoving nodes of the rope, those nodes being regions where there is no chance of finding the electron. The continuity of the rope from point to point captures a rough approximation of the differential equations that assign high energy costs to sharp bends in the rope. The absolute rotation speed of the rope represents the total mass-energy of the electron, or at least can be used that way.
Finally, and a bit more complicated, you can break those simple loops down into other wave components by using the Fourier transform. Any simple look can also be viewed as two helical waves (like whipping a hose around to free it) going in opposite directions. These two components represent the idea that a single-loop wave function actually includes helical representations of the same electron going in opposite directions, at the same time. "At the same time" is highly characteristic of quantum function in general, since such functions always contain multiple "versions" of the location and motions of the single particle that they represent. That is really what a wave function is, in fact: A summation of the simple waves that represent every likely location and momentum situation that the particle could be in.
Full quantum mechanics is far more complex than that, of course. You must work in three spatial dimensions, for one thing, and you have to deal with composite probabilities of many particles interacting. That drives you into the use of more abstract concepts such as Hilbert spaces.
But with regards to the question of "why complex instead of real?", the simple example of the similarity of quantum functions to rotating ropes still holds: All of these more complicated cases are complex because, at their heart, every point within them behaves as though it is rotating in an abstract space, in a way that keeps it synchronized with points in immediately neighboring points in space.
A: EDIT add:
My Answer is GA centric and after the comments I felt the need to say some words about the beauty of Geometric Algebra:
On 2nd page of Oersted Medal Lecture (link bellow):

(3) GA Reduces “grad, div, curl and all
  that” to a single vector derivative
  that, among other things, combines the
  standard set of four Maxwell equations
  into a single equation and provides
  new methods to solve it. 

Geometry Algebra (GA) encompasses in a single framework for all this:
Synthetic Geometry, Coordinate Geometry, Complex Variables, Quaternions, Vector Analysis, Matrix Algebra, Spinors, Tensors, Differential forms. It is one language for all physics.
Probably Schrödinger, Dirac, Pauli, etc ... would have used GA if it existed at the time.
To the Question: WHY is the wave function complex? This Answer is not helpful: because the wave function is complex (or has a i on it). We have to try something different, not written in your book.
In the abstracts I bolded the evidence that the papers are about the WHYs. If someone begs a fish I'll try to give a fishing rod.
I'm an old IT analyst who would be unemployed if I had not evolved. Physics is evolving too.
end EDIT
Recently I've found the Geometric Algebra, Grassman, Clifford, and David Hestenes.  
I will not detail here the subject of the OP because each one of us need to follow paths, find new ideas and take time to read. I will only provide some paths with part of the abstracts:  
Overview of Geometric Algebra in Physics 
Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics  (a good start)  

In this lecture Hestenes is arguing
  for a reform of the way in which
  mathematics is taught to physicists.
  He asserts that using Geometric
  Algebra will make it easier to
  understand the fundamentals of
  physics, because the mathematical
  language will be clearer and more
  uniform.

Hunting for Snarks in Quantum Mechanics 

Abstract. A long-standing debate over
  the interpretation of quantum
  mechanics has centered on the  meaning
  of Schroedinger’s wave function ψ for
  an electron. Broadly speaking, there
  are two major  opposing schools. On
  the one side, the  Copenhagen school
  (led by Bohr, Heisenberg and Pauli) 
  holds that  ψ provides a  complete
  description of a single electron
  state; hence the probability 
  interpretation of ψψ* expresses an
  irreducible uncertainty in electron
  behavior that is intrinsic in  nature.
  On the other side, the realist school
  (led by Einstein, de Broglie, Bohm and
  Jaynes) holds  that  ψ represents a
  statistical ensemble of possible
  electron states; hence it is an 
  incomplete  description of a single
  electron state. I contend that the
  debaters have overlooked crucial facts
  about  the electron revealed by Dirac
  theory. In particular, analysis of
  electron zitterbewegung (first noticed
  by Schroedinger) opens a window to
  particle substructure in quantum
  mechanics that explains the  physical
  significance of the complex phase
  factor in  ψ. This led to a  testable
  model for particle  substructure with
  surprising support by recent
  experimental evidence. If the
  explanation is upheld by  further
  research, it will resolve the debate
  in favor of the realist school. I give
  details. The perils of  research on
  the foundations of quantum mechanics
  have been foreseen by Lewis Carroll in
  The  Hunting of the Snark!

THE KINEMATIC ORIGIN OF COMPLEX WAVE FUNCTION 

Abstract. A reformulation of the Dirac
  theory reveals that i¯h has a
  geometric meaning relating it to
  electron spin. This provides the basis
  for a coherent physical interpretation
  of the Dirac and Sch¨odinger theories
  wherein the complex phase factor
  exp(−iϕ/¯h) in the wave function
  describes electron zitterbewegung, a
  localized, circular motion generating
  the electron spin and magnetic moment.
  Zitterbewegung interactions also
  generate resonances which may explain
  quantization, diﬀraction, and the
  Pauli principle.

Universal Geometric Calculus a course, and follow:
III. Implications for Quantum Mechanics

The Kinematic Origin of Complex Wave Functions
  Clifford Algebra and the Interpretation of Quantum Mechanics
  The Zitterbewegung Interpretation of Quantum Mechanics
  Quantum Mechanics from Self-Interaction
  Zitterbewegung in Radiative Processes
  On Decoupling Probability from Kinematics in Quantum Mechanics
  Zitterbewegung Modeling
  Space-Time Structure of Weak and Electromagnetic Interactions  


to keep more references together:
Geometric Algebra and its Application to Mathematical Physics (Chris Thesis)  
(what lead me to this amazing path was a paper by Joy Christian 'Disproof of Bell Theorem')
'Bon voyage', 'good journey', 'boa viagem'   
A: From the Heisenberg Uncertainty Principle, if we know a great deal about the momentum of a particle we can know very little about its position. This suggests that our mathematics should have a quantum state that corresponds to a plane wave $\psi(x)$ with a precisely known momentum but entirely unknown position.
A natural definition for the probability of finding the particle at the position $x$ is $|\psi(x)|^2$. This definition makes sense for both a real wave function and an imaginary wave function.
For a plane wave to have no position information is to imply that $|\psi(x)|$ does not depend on position and so is constant. Therefore we must have $\psi$ complex; otherwise there would be no way to store the information "what is the momentum of the particle".
So in my view, the complex nature of wave functions arises from the interaction between the necessity for (1) a probability interpretation, (2) the Heisenberg uncertainty principle, and (3) plane waves.
A: Alternative discussion by Scott Aaronson:
http://www.scottaaronson.com/democritus/lec9.html


*

*From the probability interpretation postulate,
we conclude that the time evolution operator $\hat{U}(t)$ must be unitary in order to keep the total probability to be 1 all the time. Note that the wavefunction is not necessarily complex yet.

*From the website: "Why did God go with the complex numbers and not the real numbers?
Answer: Well, if you want every unitary operation to have a square root, then you have to go to the complex numbers... " $\hat{U}(t)$ must be complex if we still want a continuous transformation. This implies a complex wavefunction.
Hence the operator should be: $\hat{U}(t) = e^{i\hat{K}t}$ for hermitian $\hat{K}$ in order to preserve the norm of the wavefunction.
A: THIS LATE ANSWER (Jan 2018) expands a little on Carl Brannen’s straightforward, and (IMO) underappreciated, answer (showing a little way above, at time of posting), which reminded me of another simple and convincing argument as to why the wave-function should be complex, set out many years ago in Dicke & Wittke's “Introduction to Quantum Mechanics” (1960; pp. 23-24).
Given their review in Ch 1 of why a quantum mechanical wave is subject to wave-particle duality/the De Broglie relation, they proceed as follows: 
For a wave-particle of sharply-defined momentum: 
λ = h/p (and thus, by Δx.Δp>= h/4π, completely uncertain – 
    essentially uniform – position) 
…the probability distribution |ψ|^2 of a plane wave should be uniform in position, which cannot be satisfied by a real-valued plane wave  
ψ = A sin(kx - ωt + α) 
…but is satisfied (generalising to an arbitrary position) by 
ψ = A exp [i (k.x- ωt)], where the propagation vector k = p/(h/2π).
Dicke & Wittke then discuss how the complex-valued wave function accounts for interference effects (Out of copyright and safely available via https://archive.org/details/IntroductionToQuantumMechanics). 
[NB Care googling the book title/pdf - many online sources, unlike the one above, are unsafe]
A: Among other things, the OP reprinted a page of a textbook, asking what "it is all about". I think it is impossible to answer this kind of questions because what is the OP's problem all about is totally undetermined, and the people who offer their answers could be writing their own textbooks, with no results.
The wave function in quantum mechanics has to be complex because the operators satisfy things like 
$$ [x,p] = xp-px = i\hbar.$$
It's the commutator defining the uncertainty principle. Because the left hand side is anti-Hermitian,
$$ (xp-px)^\dagger = p^\dagger x^\dagger - x^\dagger p^\dagger = (px-xp) = -(xp-px),$$
it follows that if it is a $c$-number, its eigenvalues have to be pure imaginary. It follows that either $x$ or $p$ or both have to have some non-real matrix elements.
Also, Schrödinger's equation
$$i\hbar\,\, {\rm d/d}t |\psi\rangle = H |\psi\rangle$$
has a factor of $i$ in it. The equivalent $i$ appears in Heisenberg's equations for the operators and in the $\exp(iS/\hbar)$ integrand of Feynman's path integral. So the amplitudes inevitably have to come out as complex numbers. That's also related to the fact that eigenstates of energy and momenta etc. have the dependence on space or time etc.
$$\exp(Et/i\hbar)$$
which is complex. A cosine wouldn't be enough because a cosine is an even function (and the sine is an odd function) so it couldn't distringuish the sign of the energy. Of course, the appearance of $i$ in the phase is related to the commutator at the beginning of this answer. See also

http://motls.blogspot.com/2010/08/why-complex-numbers-are-fundamental-in.html
Why complex numbers are fundamental in physics

Concerning the second question, in physics jargon, we choose to emphasize that a wave function is not a scalar field. A wave function is not an observable at all while a field is. Classically, the fields evolve deterministically and can be measured by one measurement - but the wave function cannot be measured. Quantum fields are operators - but the wave function is not. Moreover, the mathematical similarity of a wave function to a scalar field in 3+1 dimensions only holds for the description of one spinless particle, not for more complicated systems.
Concerning the last question, it is not useful to decompose complex numbers into real and imaginary parts exactly because "a complex number" is one number and not two numbers. In particular, if we multiply a wave function by a complex phase $\exp(i\phi)$, which is only possible if we allow the wave functions to be complex and we use the multiplication of complex numbers, physics doesn't change at all. It's the whole point of complex numbers that we deal with them as with a single entity.
A: If the wave function were real, performing a Fourier transform in time will lead to pairs of positive-negative energy eigenstates. Negative energies with no lower bounds is incompatible with stability. So, complex wave functions are needed for stability.
No, the wave function is not a field. It only looks like it for a single particle, but for N particles, it is a function in 3N+1 dimensional configuration space.
A: Already great answers to this often asked question. Very simply put tho, quantum eigenstates have associated relative phase values, and the in-phase and quadrature plane (also strangely but conventionally referred to as the Real/Imaginary plane) provides for specifying these phases when plotting or otherwise specifying a wave function. 
As pointed out in other answers, there are other mathematical ways to do this, but the “complex” way is mathematically very convenient. 
A: Since the physical point of view, the wave function needs to be complex in order to explain the double-slit experiment, as well mentionated in the book of The Feynman Lectures on Physics-III, I suggest you that review chapters 1&3, where it is explained how $\psi$ has to be considered of probabilistic nature, according to the pattern of interference, because "something" has to behave like a wave at the time of crossing through "each one" of the slits. Furthermore, Bohm proclaims that path of the particle (electron,photon, etc.) can be considered classic, so as a consequence you may watch this one, as it follows the rules already known at the macro... in that sense, you can see next reference or this one to consider the 
covariance of the laws of mechanics.
A: The wavefunction $\psi(x)$ is the projection of the physical system's state vector $|\psi\rangle$ onto the $\hat{x}$ eigenket $|x\rangle$ of eigenvalue $x$, viz. $|\psi\rangle=\int dx\psi(x)|x\rangle$. You mustn't confuse the scalar-value $\psi(x)=\langle x|\psi\rangle$ with the vector $|\psi\rangle$ that lives in a Hilbert space.
The first sentence of your first question is, in technical terms: why is this Hilbert space over the field $\mathbb{C}$ rather than, say, $\mathbb{R}$? If you Ctrl+F to "Real vs. Complex Numbers" here, you'll get a detailed discussion of several motivations for why quantum mechanics ought to look like that. One advantage of a complex wavefunction is it has both an amplitude and a phase, but only the former affects the probability density $|\psi|^2$, and the latter gives us quantum interference because of trigonometric identities such as $|1+\exp i\theta|=2|\cos\frac{\theta}{2}|$. However (to continue your Q1), a Galilean transformation needs to include a phase shift so the Schrödinger equation will be invariant; see here and here for more information. A gauge transformation such as the Galilean one is simply a way of transforming coordinates, or fields (which come to the same thing in a Lagrangian field theory), which leave the action and its equations of motion invariant. (By the way, you need to be careful not to confuse the words transform and transformation.)
Your Q2 also hinges on not confusing $\psi$ with $|\psi\rangle$. The ray is the set of values of $\exp i\theta|\psi\rangle$ with $\theta\in\mathbb{R}$, but switching from one value of $\theta$ to another leaves $\psi$ invariant because $\langle x|$ gets multiplied by $(\exp i\theta)^\ast=\exp -i\theta$.
As for Q3, it's more useful to work with the modulus and phase of the complex $\psi$ rather than its real and imaginary part, because under unitary transformations the former is invariant and so are differences in the latter.
A: Since both the amplitude and the wavelength cannot be known with precision simultaneously, I think of this as meaning that there is some missing information that must still be dealt with continuously.  That information is conveniently stored in the imaginary part of a complex number.
