Is there a direct physical interpretation for the complex wavefunction? The Schrödinger equation in non-relativistic quantum mechanics yields the time-evolution of the so-called wavefunction corresponding to the system concerned under the action of the associated Hamiltonian. And this wavefunction is, in general, complex, and its modulus squared yields the probabilities observed experimentally. Though, perhaps, this question has been asked many times, I am wondering if there is a direct physical interpretation - something that physically corresponds to - the wavefunction. Or is it just an intermediate calculational tool to arrive at the appropriate predictions for experimental outcomes, and nothing more? Of course, things like superposition and interference effects follow from the complex nature of the probability amplitude. So there must be something physical about it. What is it? Or are we not supposed to ask that question?
Is it because the probability amplitude is complex that we have difficulty in relating it to something physical? Can we do quantum mechanics without complex numbers?
 A: On your question: Can we do quantum mechanics without complex numbers?
Yes. One can in general replace any complex number by a 2x2 real valued matrix.
$a+ib ~=~\left(\begin{array}{rr} ~~a & -b \\ b & a \end{array}\right)$
Other examples are the complex Pauli matrices and the quaternions which can be both replaced by 4x4 real valued matrices. There isn't anything magical or special in the use of complex values in physics.
A: Can we do quantum mechanics without complex numbers? Yes.
Use Geometrical Algebra (GA) as a simpler framework to express physics:
Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics
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.

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.

Using Geometric Algebra an intuitive view is around the corner (geometry notions fit better in my head). 
In this PSE I link a list of resources of GA.
A: The problem is not so much that you're not supposed to ask the Question, it's more that if you ask the Question you may be swamped by many different Answers, which will have relationships between them that you may well not be able to understand unless you have already read widely.
A moderately standard Answer is that the Born interpretation of the wave function gets you a long way. You can model the statistics of experimental raw data quite nicely by the probability measures that come out of the mathematics of quantum mechanics, if you get the right model for the experimental apparatus.        One simple-minded way to say what QM predicts when measurements don't commute (that is not very standard) is that the probabilities come out negative, and you can't do an experiment that gets statistics that match those probabilities, of which we can say that those measurements are incompatible.
An off-beat justification for complex numbers —IMO, definitely not standard, and there are certainly other attempts at this— is Leon Cohen's paper "Rules of Probability in Quantum Mechanics", Foundations of Physics 18, 983(1988) (which ties probabilities to complex structure by showing that the introduction of a characteristic function approach makes a complex structure natural — though this should make you worry about circularity), which sadly is only available behind a paywall, at https://doi.org/10.1007/BF01909934, being too early for an arXiv preprint version to exist.
EDIT: But algebraic completeness is a very good reason, which has the advantage that it trips off the tongue nicely.
EDIT(2): The question is perhaps whether there is a natural complex structure. The only possible candidate, as far as I've ever seen, is the Hodge dual, in tensor form ${\epsilon^{\alpha\beta}}_{\mu\nu}$, in the exterior calculus $\star$, but so far I've not liked anything I've seen or that I've tried to construct that uses this structure. Frankly, it's not often easy to take seriously approaches that take the Hodge dual with ontological seriousness. The usual approach effectively introduces a complex structure $i$ as the imaginary that is used whenever one constructs a Fourier transform, which is a quite natural introduction, but is not for any other reason a natural structure.
A: I'll leave an answer for your last question on whether complex numbers are necessary for QM.
Scott Aaronson has a nice lecture here http://www.scottaaronson.com/democritus/lec9.html , scroll down to the section on Real vs. Complex numbers.
My favorite argument there is the first one -- that if you have a linear operator $U$, then you would want to have operators like $V$ where $V^2=U$, simply because you expect continuity; i.e. if you're allowed to do one full transformation, you should be able to do "half" of it too. (If waiting for one second is allowed, then waiting for half a second should also be allowed). In order to have square roots of operators in general, you'll need to allow operator matrices with complex elements. And once you allow that, the state vectors that they act on will also need to be complex in general. And so your wavefunction will also need to be complex.
A: The physical interpretation of a wave-function is correctly given in almost all text-books. Its being "unusual" is due to too simplified teaching the classical mechanics. For example, ask yourself what the Moon position is. It is an average of many data points. The CM certainty is obtained as a result of averaging many data points. "Many data points" is an intrinsic thing for physical phenomena. Indeed, can you convince anybody in anything if you bring only one point on your photo-film? In QM the position is not a function of time anymore but an operator with different eigenvalues. Ensemble of these eigenvalues describes a state. One point does not describe a state, unfortunately. A photo of the Moon is different from a photo of Mars in details that are different points.
So arrays of data are not unusual to physics. They are necessary and are implied in our notions of space, time, reference systems, etc. These arrays obey their own laws. These laws are sometimes wave laws. So the wave function is a representation of the data describing a given physical system if it is observed "many times". Without averaging it is more detailed than after averaging. To cope with complex waves, think of the light description in terms of complex amplitudes and of the way to get a real-valued intensity.
A: There is a difference between the physical meaning of the wave function and the physical meaning of one value of the wave function. Consider a system of two components, indistinguishable, with one spatial degree of freedom and spin one-half.  Then for example we could have psi(q_1,p_2,s_1,s_2) 
where s_i are spin variables taking the values 1,-1. {Or, choosing a different polarisation of configuration space, it could have been xi(q_2,p_1,s_1,s_2).}  (I omit the conditions which psi must satisfy.)
The physical meaning of psi as a whole (up to a phase factor) is that it is the 
"state"
 which the system is in.  "State" is a physical term, it includes all physical properties of 
the system.  Any further question such as 'what is a state' is essentially philosophy, 
not physics.
The physical meaning of the value of psi (provided psi is normalised to have L^2 norm unity) at a particular value of q_1, p_2, s_1, and s_2, is that the square of the modulus of the value is the probability that the system will yield a measurement result of position of first compononet = q_1, momentum of second component = p_2, spin of first component = s_1, spin of second component = s_2  --- provided, of course that the system interacts with the appropriate measurement apparatus for this set of questions.  
The physical meaning of psi as a whole, or of xi as a whole, is the same.  And this meaning is simply one of the six axioms of QM.  The physical meaning of the values of psi is different from those of xi, but these physical meanings follow logically from the physical meaning of psi or xi as a whole plus the axioms of measurement plus the definitions of the spin observables, position observables, and momentum observables.  
Just as one could study a function without picking coordinates (and hence a fortiori no picking a polarisation of configuration space), and without studying its values, so the physical meaning of psi makes sense independently of the physical meaning of its values, and is, in the usual axiomatic framework of QM, logically prior.  But there are re-constructions of QM which reverse this order.  Some people prefer those re-constructions....Lucien Hardy is the most famous such re-constructor, and has attempted it twice (his system gets more and more convoluted each time....) 
The post by Vladimir Kalitvianski is very sensible: the values of psi are, indeed, a set of measurable data, and a suitably chosen 'array' of them suffices to determine psi completely (up to a phase factor).
One cannot use similar real-valued functions, because phase relations are physical.  If one tried to use real-valued functions only, it would not describe all the physical properties of the system (it could not take into account the phase relations).
A: After many years thought devoted to what the wave function represents, I have come to the conclusion that is it a complex-valued probability distribution over configuration space, and not the representation of the physical state of the system as usually claimed. Because of the uncertainty principle we cannot know all properties of a system, but only some. When we have maximal possible knowledge (i.e. when we are in a pure state of knowledge) we know some properties (e.g. a particle's momentum) and quantum theory comes up with a probability distribution over other properties (e.g. over the particle's position in space). We can thus consider the quantum mechanical formalism as a theory of complex-valued probability and it seems we can construct rules of such a probability theory more or less independently of that formalism, and include among these rules principles for assigning probabilities in a Bayesian manner. (These are generalisations of the principle of indifference, the method of transformation groups and the maximum entropy principle, as formulated, for example, by Jaynes in the case of real-valued (non-negative) probabilities.) Without contradicting the usual quantum mechanical formalism, we can thus adopt a realist approach to the interpretation of quantum theory. 
If you want to see my work on this you will find reference to it in my LinkedIn profile.
