Why is quantum physics based on complex numbers?

The modern quantum theory is based on the assumption that quantum states are represented by elements of a complex Hilbert space. It is expected that in future quantum theory the number field will not be postulated but derived from more general principles. We consider the choice of the number field in a quantum theory based on a finite field. We assume that the symmetry algebra is the finite field analog of the de Sitter algebra so(1,4) and consider spinless irreducible representations of this algebra. It is shown that the finite field analog of complex numbers is the minimal extension of the residue field modulo p for which the representations are fully decomposable.