Why are Only Real Things Measurable?

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.

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$^1$ 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 $C=\frac{A-A^{\dagger}}{2i}$.

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.

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.