Oh yeah, Veritasium has some great visualizations of that too. I cant remember what the video was called but it was about fractals and the Newton Raphson method, and he had these input/output complex planes left and right to illustrate it.
Again why would anyone posit that a pair of numbers suddenly appears when trying to solve x^2+k=0 as k goes from negative to positive?
And why is two dimensions enough for third and higher degree equations? Were there a simpler geometric connection, you’d probably have a nicer proof of the Jordan curve theorem… but you don’t.
Not sure what you mean by pair of pairs… the OP said that calling imaginary numbers “double numbers” would clear up a lot of issues but that is not clear at all to me.
By pair of pairs I just mean that you're looking for two solutions, and they turn out to be computed numbers, both of them.