The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that's used for geometry or 2D coordinates or anything.

Then you progress into all the complex e^ functions where you basically forget about a plane, and it's just a convenient shorthand for math that would otherwise involve a bunch of trigonometric functions. But again, this never provides any actual intuition... it's just convenience. You could still write everything as sin() and cos() etc.

I finally felt like I understood complex numbers when I asked myself, how do you create a continuous solution to the function y(x) = (-1)^x? Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...]. But you can work out the math such that it necessarily produces a spiral through them. And it's different from the e^ equations because there's no pi involved.

And so rather than thinking of complex numbers as a "plane", it's much better to think of them simply as the inherently spiral motion required to continuously join alternating values. And when I look at how complex numbers are actually used in things with physical correlates -- e.g. signal processing -- the spiral intuition always continues to make sense.

Of course, at the end of the day, it's all the same math. But the idea of a continuous spiral path between otherwise discontinuous real numbers has wound up clicking for me in a way that the metaphors of a 2D plane, or of more generalized arbitrary "rotation", never has. That complex numbers are about spiral oscillation, not about planes.