There is a fundamental theorem already conjectured by von Neumann but proved just at the end of 1900 by Solèr (in addition to a partial result already obtained by 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 $L^2$ space.
The second possibility is the one considered "standard" nowadays, a Hilbert space over the field of complex numbers. The third, quite exotic possibility, is a Hilbert space whose scalars are quaternions. This third possibility has been investigated by several authors (see the book by Adler for instance).
It is possible to prove, however, that the first possibility is only theoretical for physical reasons. In fact when one deals with physical systems described in real Hilbert spaces, invariant under time-reversal, the time reversal operation induces a complex structure and, in fact it is equivalent to deal with a complex Hilbert space.