There is a fundamental result already conjectured by von Neumann but proved just at the end 20'th century by Solèr (in addition to a partial result already obtained by Piron in the sixties) which establishes (relying on the theory of orthomodular lattices and projective geometry) that the general phenomenology of Quantum Mechanics can be described only by means of three types of Hilbert spaces.
(All fundamental theorems of quantum theory like, say, Wigner theorem, can be proved in these three cases.)
One is a Hilbert space over the field of real numbers.
In this case wavefunctions can be taken as real valued functions if the system is described in terms of a $L^2$.
The pure states are here unit vectors up to signs instead of up to phases.
It is therefore evident that decomposing the complex wavefunction into real and complex part is not a real QM since in that case pure states are unit vectors up to $SO(3)$ rotations.
The second possibility is the one considered "standard" nowadays, a Hilbert space over the field of complex numbers and the pure states are unit vectors up to phases as is well known.
The third, quite exotic possibility, is a Hilbert space whose scalars are quaternions. Now pure states are unit vectors up to quaternionic phases (quaternionic factors with unitary norm).
This third possibility has been investigated by several authors (see the book by Adler for instance).
As a matter of fact we only know physical systems described in complex Hilbert spaces, is there a fundamental reason to rule out the other two cases?
It seems possible to prove that the first possibility is only theoretical for physical reasons under some hypotheses. When one deals with physical systems described in real Hilbert spaces CCRs induce a complex structure and, in fact it is equivalent to deal with a complex Hilbert space.
CCRs of $X$ and $P$ can enter the game through two different ways. You can suppose that your theory admits position and momentum as fundamental operators, or you may assume that your system is covariant under the action of a unitary irreducible representation of the (extended) Galileo group. I find both possibilities not very satisfactory. On the one hand position is not so relevant (massles particles do not admit any notion of position operator), on the other hand Galileo's group is not fundamental in modern physics. Together with a PhD student of mine, M. Oppio, I recently discovered a much more satisfactory argument (http://xxx.lanl.gov/abs/1611.09029 published in Reviews in Mathematical Physics 29 n.6, (2017) 1750021 DOI: 10.1142/S0129055X17500210) to introduce a natural unique (Poincare' invariant) complex structure in a quantum theory initially formulated in a real Hilbert space, assuming to deal with an elementary system of relativistic nature, supporting an irreducible representation of Poincaré group in terms of automorphisms of the lattice of elementary propositions and admitting an irreducible von Neumann algebra of observables generated by the representation itself (within a refinement of Wigner's idea of elementary relativistic system).