You can literally take a class on Complex Analysis. Turns out that those "small" modifications have huge ramifications. They add a ton of extra structure to the real numbers which can be exploited, particularly if your problem can be expressed in terms of sines and cosines, or if your problem lives on a plane.

For example, complex differentiability is much more stringent than real differentiability, to the point that the existence of one complex derivative implies the existence of all of them! Furthermore, you have to be really careful extending the classic functions to the complex numbers. Typically, you either end up with a multivalued function, or you have to pick a specific branch that is single-valued.

If you want to learn more, Theodore Gamelin's Complex Analysis book is a good place to start. But to read it, you'd really benefit from a background in vector calculus. For a more "practical" but still detailed account of complex variables, check out Complex Variables and the Laplace Transform for Engineers by Wilbur LePage, which just assumes basic calculus.

Is there much beyond i^2 = -1, z = a + bi, and e^iθ = cosθ + isinθ

What does electric current "i" have to do with the glorious imaginary unit j ?

This post was brought to you by Electrical Engineering Gang.

!In electrical engineering, we use the letter j instead of i because historically, i is reserved for electrical current. Gamelin uses i in his book ~~which is wrongthink~~, but LePage uses j. !<