>C is twisted and weird.

Why do you say this? I've admittedly never done a proper complex analysis course but I got the impression that that complex differentiability was a very strong condition that results in holomprhic functions behaving "nicely" in ways that real functions do not

Should have used quotes.. I didn't mean it in any formal sense. What I am saying the nature of unit in complex plane makes it difficult to intuitively imagine invertibility and determinants.