Complex numbers behave quite differently than vectors/matrixes, and their discovery wasn't motivated by needing to represent coordinates. They were discovered (or invented, for the nonplatonists) in the process of resolving the paradox that 3rd degree polynomials must have at least 1 (real) root (because they always stretch out in opposite directions, and so cross 0), but you end up needing to square root a negative number to solve for it (the paradox being, this was thought to be impossible, since the square of any real number is positive).

Their existence is a natural consequence of the rules of the algebraic operations (addition, subtraction, multiplication, division, powers and logarithms); they aren't incidental, they are integral. Without imaginary numbers, the algebraic operations aren't closed, and so don't form an algebra.

(This being a brief summary of the thesis of the videos I linked in a sibling comment.)

That being said, they are useful as coordinates, as well.