  In standard quantum mechanics, complex numbers are used to describe the
wavefunction. Although complex numbers have proven sufficient to predict the
results of existing experiments, there is no apparent theoretical reason to
choose them over real numbers or generalizations of complex numbers, i.e.
hyper-complex numbers. Experiments performed to date have proven that real
numbers are insufficient, but whether or not hyper-complex numbers are required
remains an open question. Quantum theories based on hyper-complex numbers are
one example of a post-quantum theory, which must be put on a firm experimental
foundation. Here we experimentally probe hyper-complex quantum theories, by
studying one of their deviations from complex quantum theory: the
non-commutativity of phases. We do so by passing single photons through a
Sagnac interferometer containing two physically different phases, having
refractive indices of opposite sign. By showing that the phases commute with
high precision, we place limits on a particular prediction of hyper-complex
quantum theories.
