Thank you for bringing up complex differentiability. Complex analysis is the only field I've found that seems to genuinely require complex numbers rather than

Thank you for bringing up complex differentiability. Complex analysis is the only field I've found that seems to genuinely require complex numbers rather than just polar coordinate notation. However I get the impression that complex analysis, as a research field, has been a dead end for a long time now with few people working on it [1]. If pure research has come to a halt then it's fair to judge the field based on the practical applications it has achieved. Are there many? The class of holomorphic functions is pretty rigidly constrained -- infinitely differentiable, equivalent to a Taylor series at each point. I once heard that even physicists doubt the legitimacy of assuming that all physical functions are holomorphic (but now I'm speaking beyond my expertise). Continuing to speak beyond my expertise, I don't think Feynman had holomorphic functions in mind when he said "our jewel". I have to say that the definition of limits for complex numbers always seemed like an artificial straitjacket. I understand why you can't have complex limits defined any other way, but maybe the take-away here isn't "complex limits should work like this" so much as "complex limits, and therefore complex analysis, is not very useful". [1] "Complex analysis is pretty much dead. There are some remaining problems but most of them are either too specialised or too difficult. This is not a very healthy situation, and there are relatively few people working in complex analysis nowadays.", Dmitry Beliaev , former Assistant Professor (Mathematics) at Princeton University, https://www.quora.com/Are-there-any-mathematics-faculty-at-Princeton-specializing-in-complex-analysis?share=1 https://www.quora.com/Are-there-any-mathematics-faculty-at-P...