Why are Only Real Things Measurable? Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with operators in quantum mechanics, why must measurable quantities be Hermitian, and consequently real?
 A: I) Well, one can identify a complex-valued observable with a normal operator 
$$\tag{1} A^{\dagger}A~=~AA^{\dagger}.$$ 
A version$^1$ of the spectral theorem states that an operator $A$ is orthonormally diagonalizable iff $A$ is a normal operator. 
Thus normal operators are the only kind of operators that we can consistently extract measurements [i.e. eigenstates and (possibly complex) eigenvalues] from.
II) But notice that a normal operator 
$$\tag{2} A~=~B+iC$$ 
can uniquely$^2$ be written as a sum of two commuting self-adjoint operators 
$$\tag{3} B^{\dagger}~=~B, \qquad C^{\dagger}~=~C,  \qquad [B,C]~=~0.  $$
($B$ and $C$ are the operator analogue of decomposing a complex number $z=x+iy\in\mathbb{C}$ in real and imaginary part $x,y\in\mathbb{R}$.) Conversely, two commuting self-adjoint operators $B$ and $C$ can be packed into a normal operator (2). We stress that the commutativity of $B$ and $C$ precisely encodes the normality condition (1). 
Since the self-adjoint operators $B$ and $C$ commute, they can be orthonormally diagonalized simultaneously, i.e. the corresponding pair $(B,C)$ of real-valued observables may be measured simultaneously. This fact is consistent with the Heisenberg uncertainty principle applied to the operators $B$ and $C$.
We conclude that a normal operator does not lead to anything fundamentally new which couldn't have been covered by a commuting pair of standard real-valued observables, i.e. self-adjoint operators. For this reason, the possibility to use normal operators as complex observables is rarely mentioned when discussing the postulates of quantum mechanics.
For more on real-valued observables, see e.g. this Phys.SE post and links therein.
--
$^1$ We will ignore subtleties with unbounded operators, domains, selfadjoint extensions, etc., in this answer.
$^2$ The unique formulas are $B=\frac{A+A^{\dagger}}{2}$ and $C=\frac{A-A^{\dagger}}{2i}$.
A: Since the complex numbers are quite abstract term, and do not have physical representation, it is still possible to see them, although only imaginary part or only real part of some measurement is not going to give full information. Only full complex number represents a full information.
Now, I said it is possible to really measure imaginary and real values. While it is not quantum mechanics, QA modulation is a good example how you can really measure imaginary and real parts of a signal.
A: At least complex energy and imaginary time are used in quantum mechanics. Complex energy to describe non-stationary processes. Imaginary time is used in Landau Lifshits’s book, Volume 3, Problem 3 to paragraph 77. At the same time, such words are used: “The imaginary value of a moment in time $\tau_0$ corresponds to the classical impracticability of the process”
 $$W=\exp\left[-2\mathrm{Im}\left(\int_{\tau}^{\tau_0}\frac{4F^2}{\Omega^2}\sin^2\Omega u\mathrm{d}u+\tau_0\right)\right]$$ 
But to use the complex eigenvalues of quantum mechanics operators, it is necessary to use non-self-adjoint operators, and then the eigenvalues can turn out to be complex. In complex space, the energy and momentum operators are general operators, non-self-adjoint.   $$\hat H=\sum_{k=1}^3 -\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial z_k^2},z_k=\mathrm{Re}\,z_k+i \mathrm{Im}\,z_k$$ $$\hat p_r=-i\hbar(\frac{\partial }{\partial r}+\frac{1}{r})$$ The eigenfunction of the radial part of the momentum operator in three-dimensional space is equal to $\psi=exp(ip_r r/\hbar)/r$.  The eigenvalues of these operators can be complex. When writing a solution for the wave function in the complex plane there is a problem. When using real space, there is a non-damping solution only with real coordinates. Similarly, in a complex plane with a complex eigenvalue, a non-damping solution exists at a certain phase of the complex coordinate. We need to think about the physical meaning of the complex solution. In hydrodynamics, the physical meaning of the imaginary part is the standard deviation. In quantum mechanics, apparently also. You need to measure the constant term described by the real part and the variable one, the fading away term described by the imaginary part. Complex energy and momentum describes the localization in time and in space, respectively, energy and momentum. The imaginary part of the complex energy value is determined from the lifetime of the system. The imaginary part of the impulse is determined from the known complex value of the energy  and equations $E^2=p^2c^2+m^2c^4$.
A: As a mathematical structure, the field of complex numbers does not admit an order relation which is an extension of the order we have in $\mathbb{R}$.
This means that there is absolutely no way of saying if $5+3i$ is bigger or smaller than $5+6i$ for example. We just know it is not equal and we have to stop here.
Therefore it is physically really hard (actually impossible) to compare "observables" having as eigenvalues complex numbers.
We could not tell anymore which particle has a bigger mass, a smaller energy and so on.
I believe that taking the real field as the primary field in which the measure results take values is just a matter of convenience. You could try to create a sort of quantum mechanics with complex eigenvalues, but then you could not fit experiments anymore and you model turns extremely less predictive.
Anyways, I have read in The Road to Reality by Penrose that some physicists considered as numerical fields somethinig like the cyclic $\mathbb{Z}_p$ with $p$ prime and extremely big.
As it is not clear if this can lead to new physics, we just stick with $\mathbb{R}$.
That's it, as far as I understand the problem.
A: Imaginary numbers can be represented as pairs of real numbers. You can also make a device, which mixes the measurement outcomes of two reals on hardware level to produce complex "amplitude" and "phase" as outcomes, which you further might call as measuring a complex number.
More generally, any measurement is eventually reading off the values on the indicators of your instruments. Those are numbers, hence reals. However, they can also be sets (arrays) of reals, as is the case for cameras, for example. So, perhaps, most general statement would be that one can measure quantities, which are expressible as a set of real numbers.
A: Well "imaginary numbers" are just that; imaginary.   We made them up to allow for the square root of a negative number.   Likewise, we made up complex numbers.
The problem is NOT with the imaginary, or complex number.   The difficulty is with the interpretation that we CHOOSE to use for an imaginary or complex number.
For example in AC electrical circuit problems,  we consider multiplying by -1 to be the equivalent of a 180 degree rotation of a vector that represents say a Voltage.  It is then just a small jump to suggest multiplying by the square root of -1 (i or j) to be equivalent to rotating the vector through half of 180, or 90 degrees, since multiplying twice by i (or j) gives us the same rotation of twice 90 degrees or 180 degrees.
So the mathematics, is nothing more than a convention for describing a 90 degree vector rotation, or a combination of two vectors at right angles.   It's no more mysterious than the Germans using a capital letter for ALL nouns.  It's just the way they do it.
So the REALITY of imaginary, or complex numbers, is nothing more than our definition of how WE HUMANS interpret them or their use.  There is NO universal mysterious reason at all.
A: Actually you can measure imaginary numbers by measuring separately the real and the imaginary part. However this is only possible in classical mechanics. In quantum mechanics measuring the two parts simultaneously is not possible, because the first measurement would necessarily change the outcome of the second measurement, as Dirac explains nicely in his book:

"One might think one could measure a complex dynamical variable by
  measuring separately its real and pure imaginary parts. But this would
  involve two measurements or two observations, which would be alright
  in classical mechanics, but would not do in quantum mechanics, where
  two observations in general interfere with one another - it is not in
  general permissible to consider that two observations can be made
  exactly simultaneously, and if they are made in quick succession the
  first will usually disturb the state of the system and introduce an
  indeterminancy that will affect the second." (P.A.M Dirac, The principles of quantum mechanics, §10, p.35)

