A feature of quantum mechanics that distinguishes it from classical Newtonian mechanics is that momentum and position are described by non-commuting operators on a Hilbert space. It has long been known that certain diffusion theories provide stochastic models for Schrödinger's equation, together with a natural way to understand this non-commutative structure. This theory is revisited here. It suggests that the origins of quantum theory might be found in an algebraic extension of the geometry of space-time to complex numbers. It is a possible stepping stone to an emergent explanation for quantum mechanics and to a unification of classical and quantum physics.