I wonder if it's possible to discover another version of quantum theory that doesn't depend on complex numbers. We may discover a formulation of quantum mechanics using p-adic numbers, quaternions or a finite field etc. Also, physical states lives on a Hilbert space. What if we consider the infinite dimensional Hilbert space to be the tangent space of an infinite dimensional manifold at some point? Is it possible to make these generalized theories and if it's possible can it lead to new predictions or resolve some of the difficulties that are present?
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