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Complex analysis is REALLY interesting imo. Real analysis can get kinda clunky in that differentiability, integrability, and analyticity, etc. are all separate things, but when you extend functions to the complex plane they all become synonymous. There are a lot of cool things like that. I don't know what parts of it would be useful for ChemE (besides the process control aspect), but it's been pretty invaluable for me in understanding Laplace and Fourier transforms and linear systems theory.
Real analysis is also good, but you don't get to the good stuff until much later, like with measure theory, Lp spaces, etc. (even then, only useful if you're into theory, like me!) Complex analysis will probably seem more useful right off the bat.
One would think, but the opposite is often true. People who really starved and later on doing well in business have sometimes the "didn't get any help" pride...