Complex time evolution, non-uniqueness of quantization and coherent state transforms
The construction of a fully consistent theory of quantum gravity remains one of the most challenging open problems in contemporary fundamental physics. In Loop Quantum Gravity, for example, one considers a one parameter family of quantizations. Contrary to common belief different quantizations of even the simplest systems with one degree of freedom may lead to physically inequivalent results with e.g. different spectra of observables.
By letting time to be complex one obtains a method of relating different quantizations which uses the natural geometry on the space of Kahler geometries and Segal-Bargmann-Hall (generalized) coherent state transforms.