Why hermitian, after all? This question is going to look a lot like a duplicate, but I've read dozens of related posts and they don't touch the subject. Here we go.
Why are observables represented by hermitian operators?


*

*Because then we'll measure real stuff, since the eigenvalues are real;

*Because hermitian operators (actually normal operators) have a complete set of eigenvectors, and we'll be sure the accessible states of our experiment will be modelled consistently;
Those are the two main reasons for the Physicists of the early XX century to have chosen hermitian operators as a mathematical model of what we can observe: mathematics proves they provide both the needed properties. 
My question is of philosophical inclination: which one of the above requirements is more fundamental? I'm inclined to think the most important property is to be able to represent the system by the right model, which is II). Real eigenvalues come as a bonus. I've read that having real eigenvalues is more important, but I don't see why they couldn't be complex... Maybe Nature is tricking us some way and we only measure their real part. The problem for me would be describing the system with a model that doesn't mirror our accessible states, and is therefore flawed.
Maybe both are fundamental on the same level. Maybe the question is stupid. What do you think?
 A: When in doubt go back to the masters. From Dirac's Principles of QM

When  we make an Observation we measure  some dynamical variable.
  It is obvious physically that the result of such a measurement must
  always  be a real number, so we should expect that any dynamical. variable that  we can measure must be a real dynamical variable.
  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 all right 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 indeterminacy that will  affect the  second. We
  therefore  have to restrict the dynamical variables that we can
  measure to be real..

According to Dirac property I in your question is more fundamental. Since Hermitian matrices are normal property II is satisfied. He also explains why a real dynamical variable should be a Hermitian matrix by starting from simple assumptions in his book. The book is quite old and is available for free.
A: Hermitian operators (or more correctly in the infinite dimensional case, self-adjoint operators) are used not because measurements must use real numbers, but rather because we almost always decide to use real numbers.
As the OP mentions at one point, you might choose to use complex numbers to label a two-dimensional screen, and in that case you'll be able to use a so-called normal operator to represent the 2-dimensional observable. (Contrary to what Dirac thought, nothing whatsoever goes wrong here.)
It should not be too hard then to accept my next claim: You can use whatever measurement scale you want to measure a quantum observable! You can label pointer positions with items of fruit if you want to, and you can still build a perfectly legitimate observable.
There is no question that the reals and complexes have enormous advantages over other more arbitrary measurement scales (due to their rich internal structure which we capitalize on in the functional analysis), but the idea that real numbers are somehow endowed with a prestigious metaphysical status is baloney.
How to define an observable with any measurement scale you want to
Step 1. Set up a bunch of particle detectors
Step 2. Attach a label to each detector
The set of labels we'll denote by $\Omega$. Examples include: $$\Omega = \{0,1\},\mathbb{R},\mathbb{C},\{\heartsuit,\clubsuit,\diamondsuit,\spadesuit\}$$
Step 3. Write out the list of all possible events
By event, I mean a subset $\Delta$ of $\Omega$ that represents a possible question like "Did a detector in $\Delta$ fire?". We'll label the event structure $\Sigma$, e.g.
$$\Sigma = \{\emptyset,\heartsuit,\clubsuit,\diamondsuit,\spadesuit,\heartsuit\clubsuit,\heartsuit\diamondsuit, \cdots,\heartsuit\clubsuit\diamondsuit\spadesuit\}$$
Step 4. Associate each event in $\Sigma$ with a projection operator
This is the hard bit, and there's no recipe for it. But you have to make sure that the family of projectors forms a Boolean algebra that perfectly mirrors the natural algebra of $\Sigma$.
We'll call the association $\sigma$, so that $\sigma:\Sigma\to\mathscr{P}(\mathscr{H})$.
And that's basically it! The object $\sigma$ (technically, a Projection Valued Measure on $\langle\Omega,\Sigma\rangle$) is a quantum observable. It contains all the probabilistic information you need to calculate the probability measure on your chosen measurement scale for any quantum state.
For example, suppose the state of the system is $\rho$ and you want the probability that a detector $\Delta\in\Sigma$ fires. The desired probability is just $p=tr[\rho\sigma(\Delta)]$.
What the heck does this have to do with Self-Adjoint operators?
Are you ready for the climax? Here it is...
IF you choose to use the measurement scale $\langle\mathbb{R},\mathscr{B}(\mathbb{R})\rangle$, THEN you will be able to build a self-adjoint operator which is precisely equivalent (in terms of the information it stores) to the PVM you constructed.
IF you choose a detection screen calibrated by $\langle\mathbb{C},\mathscr{B}(\mathbb{C})\rangle$, THEN repeat the above sentence replacing 'self-adjoint' with 'normal'.
(Neither of the above statements is obvious, by the way. They are famous results in Functional Analysis known as the Spectral Theorems.)
IF you choose to be a Fancy Nancy and use $\{\heartsuit,\clubsuit,\diamondsuit,\spadesuit\}$ for a measurement scale (with its power set for the event structure) THEN the fruits of your labors are more modest. In particular, you still get the answers to any questions you care to ask, but you don't get any neat operator to give you computational shortcuts. Instead you will forever be doing calculations like $p=tr[\rho\sigma(\heartsuit\clubsuit)]$.
I haven't even touched eigenvectors yet, but suffice it to say that they also do not have a fundamental status in the theory.
There is no doubt that we can learn something by reading the works of the great masters, but taking that work as the state of play can send you back a century. We've learned a lot since Einstein and Dirac.
