I'm an engineering-physics student taking a course in complex analysis, and it's a little frustrating, because I see all these connections to vector calculus over the reals (especially as applied to electromagnetics). For example, if we treat complex numbers like vectors, then the Cauchy-Riemann equations tell us that the derivative exists if the divergence and curl are both zero, which is equivalent to saying that our field is conservative, and there's no field sources/sinks. That these functions also satisfy Laplace's equation is also fairly obvious, since Laplace's equation describes a conservative field with no field sources.
Essentially, to what extent can we pretend the complex numbers are vectors, and bring over useful results from vector calculus? If this doesn't work, then what are the major flaws, or things to watch out for?
Edit: The reason I'm looking for analogs between vector analysis and complex analysis is because results in vector analysis usually have clear physical interpretations. I'm hoping to enhance my understanding of complex analysis through the similarities. Additionally, if I know a proof for certain proprerties of vector functions, I probably shouldn't focus my attention on the proof of an equivalent property in complex analysis. Instead, I'd prefer to focus attention on results that don't have a correspondence to vector analysis.
So I'm wondering, what are some results in complex analysis that don't have a similar result in vector analysis? Or even for results that are similar, what are the special peculiarities to complex analysis that make the results different?