  Quantum mechanics has withstood every experimental test thus far. However, it
relies on ad-hoc postulates which require experimental verification. Over the
past decade there has been a great deal of research testing these postulates,
with numerous tests of Born's rule for determining probabilities and the
complex nature of the Hilbert space being carried out. Although these tests are
yet to reveal any significant deviation from textbook quantum theory, it
remains important to conduct such tests in different configurations and using
different quantum states. Here we perform the first such test using coherent
states of light in a three-arm interferometer combined with homodyne detection.
Our proposed configuration requires additional assumptions, but allows us to
use quantum states which exist in a larger Hilbert space compared to previous
tests. For testing Born's rule, we find that the third order interference is
bounded to be $\kappa$ = 0.002 $\pm$ 0.004 and for testing whether quantum
mechanics is complex or not we find a Peres parameter of F = 1.0000 $\pm$
0.0003 (F = 1 corresponds to the expected complex quantum mechanics). We also
use our experiment to test Glauber's theory of optical coherence.
