Complexifying the Complex

I’ve been working recently on trying to understand exactly how Wick rotation works in two-dimensional conformal field theory. There is an analog there of some of the issues with Wick rotation of spinors and twistors in four dimensions that I’ve been struggling with. One of the few times I’ve tried to get help from an AI agent, I asked it to point me to discussions of this topic in the literature. It gave me a bunch of suggestions I already knew about, then emphasized that the two best places were a certain textbook and another set of lecture notes. Since I hadn’t heard of these, this was exciting news, so I immediately went to look them up. Hallucinations, they don’t exist. In what I have been reading, I’ve repeatedly come up against what has become a pet peeve, related to the second pet peeve discussed here. In the 2d conformal field theory literature, one is continually told that one is dealing with functions of $z$ and $\overline z$, but $z$ is independent of $\overline z$.…