Are the basic postulates of QM, such as complex Hilbert space, unitary evolution, Hermitian operator observables, projection hypothesis etc., the unique and only set of postulates that gives rise to a semi-deterministic and semi-probabilistic theory, in which the time evolution is non-degenerate? By non-degenerate, I mean different initial states never produce the same final state probabilities, which in QM is guaranteed by unitarity.
Phrased in another way, is it possible to prove from some general principles, such as semi-determinism, semi-reversibility (not for collapse), causality, existence of non-compatible observables etc., that a physical theory must satisfy these postulates? In particular, is it possible to prove that complex numbers, or a mathematical equivalent, must be fundamental to the theory?
I haven't studied anything about foundational issues of QM, so feel free to point out if I'm being a crackpot. I suppose this question may be similar to something like "can you prove that gravity must be a metric theory entirely from the equivalence principle?", whose answer is no, but I'll be glad if it turns out to be otherwise.