When should I care about complex numbers?

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I'm a very slow learner, it can be years after I "learned" something that it finally clicks. I have done exercises on complex numbers, gone to lectures, passed exams, watched a bunch of videos; Very little of it has stuck and I have never used them for anything useful.

I think I've had insight recently. Below is a comprehensive list of everything I know about complex numbers

1) They're not Real
2) They exist because there needs to be a way to find the roots of all polynomials
3) There is geometry involved. A function in 2 dimensional real space produces a line, in 2D complex space it's a plane. Or something

How to be less retarded?