The complex numbers is just the ring such that there is an element where the element multiplied by itself is the inverse of the multiplicative identity. There are many such structures in the universe.

For example, reflections and chiral chemical structures. Rotations as well.

It turns out all things that rotate behave the same, which is what the complex numbers can describe.

Polynomial equations happen to be something where a rotation in an orthogonal dimension leaves new answers.