Are there any well-developed (string?..) theories assuming that, what we perceive as a (3+1) Minkowskian manifold, is a projection/compactification of a 5-dim spacetime, locally obtained via complexification of the usual time? And this additional imaginary time direction is also the one we use for Wick rotation in QFT 🙃
Complex time has been studied and has a name ("spacekime"). There's a whole group who study it, their web page is here. I'm not sure if they've looked at connections with QFT though, as I recall they've mostly done classical theories... but I've only briefly looked at their work, I may have missed something.