The complex plane is adequate because there is a theorem that says so. It's called The Fundamental Theorem Of Algebra, and you shouldn't have any trouble finding lots of information about it in books and online. Warning: the proof takes a bit of heavy lifting.
EDIT: For a polynomial equation in two variables, even something as simple as $x-y=0$, there will generally be "points at infinity", so you need to go to projective space for a full understanding of the solutions. In one variable, only finitely many solutions, no points at infinity, no need to projectivize.