Many people have considered alternatives to standard quantum mechanics in which the Hilbert space is over the real instead of the complex numbers - see e.g. here, here, here, here, here, and here. In general, this alternate theory is surprisingly similar to standard quantum mechanics over the complex numbers:
All the greatest hits are still there: interference, entanglement, Bell inequality violations, noncommuting observables, non-unique decompositions of mixed states, universal quantum computing, the Zeno effect, the Gleason and Kochen-Specker theorems.
But I'm not sure how one would generalize the Schrodinger equation to the real setting. In order to get a unitary time-evolution operator, you need to exponentiate an anti-Hermitian operator. The only operator conveniently lying around is the Hamiltonian, which is Hermitian. Fortunately, with the complex numbers there's an easy way to convert a Hermitian operator into an anti-Hermitian one: you just multiply it by $-i$, as is done in the Schrodinger equation. But with the real numbers, I can't think of any natural way to convert the symmetric Hamiltonian operator into an antisymmetric operator which exponentiates to an orthogonal time-evolution operator. How would this work?