In a deterministic classical theory we describe the positions and velocities of particles using real numbers.
In a non-deterministic classical theory we describe the positions and velocities of particles using real random variables.
In quantum theory we can describe positions and velocities of particles using elements of $C^*$-algebras. But $C^*$-algebras are algebras over complex numbers.
Is there a reason we can't generalize real classical random variables to real quantum random variables?
