I know, it seemed bizarre even in high school precalculus. What a mess!
The graphs and trigonometric functions are necessary because the complex numbers by convention exists on a plane outside the real number line, and their values can be represented by angles and trig functions. I never took linear algebra, so I don't understand how the matrix represents i.
There are algebraic polynomials that do not have solutions using the real numbers from the traditional real number line, they only have solutions that lie on the complex numbers plane. That's why the universe of numbers extends into a two dimensional plane off of the real number line.
The thing that interests me is the question about whether these imaginary numbers are just conveniences that simplify calculations, or whether they have an independent connection to reality.
It's at least as interesting a question as transgender athletics and drag queen shows, but maybe that's just me!