You touch several interesting points here. At some time I worked on the holomorphicity of functions in the context of QM when showing its unitarity and I believe it had to do with the Stone-von Neumann theorem and the unitary map it allows in Bargmann-Segal space and its transforms for adjoint operators but it's been a while and don't recall all the details. When extending to relativistic quantum fields the Stone-von Neumann no longer holds and indeed one has to separate positive and negative energies with a procedure that involves complexification and (for the interacting case) analytic continuation to determine iepsilon prescription of Feynman propagators(LSZ normal ordering and all that stuff). And then one has not only commuting but anticommuting operators (i.e. we have both holomorphicity and antiholomorphicity) unlike in nonrelativistic quantum mechanics.As for QT, the case for the necessity of complex numbers seems to be much clearer than the case for complex numbers in SR. Orthodox QM is patently laden with complex numbers, complex vector spaces and holomorphic functions. There are tonnes of equations in QFT with explicit complex notions such as the gamma function.
The positive frequency condition of QFT - i.e. positive energy - can be deeply demonstrated to be dependent on the underlying complex structure. Moreover, the scale invariance of renormalization group theory is also deeply dependent on the notion of complex numbers.
So I would say that somewhat subtly there is a clear need for complex objects at least when dealing with more than one particle or beyond the semiclassic approximmation since in the latter one never has to mix the holomorphic unitarity of the Schrodinger evolution with the non-holomorphicity of quantum measurement using the Born rule.
Hope I didn't introduce more confusion, maybe @Auto-Didact can further comment on these points.