I find it curious that Complex Numbers give enough flexibility to be algebraically closed, where the reals, rational numbers do not. For the reals it is easy to see that they cannot be used to solve equations like $x^2 + 1 =0$. Geometrically, one can look at the number line as see that any $x$ squared yields a positive number which when added to one cannot get you back to zero. In the complex case, however, we are working with the plane. In this case exponents stretch and rotate any given $x$. It is easy to therefore see in the particular circumstance that if $x=i$ that $x^2$ rotates it to $-1$ which when added to one yields the desired result (i.e. $0$). So because the Complex Numbers are algebraically closed, I conclude that any polynomial equation with complex coefficients my be solved by choosing one or more $x$'s in the plane and rotating them and stretching them such that they will combine using the given coefficients to produce the RHS.
Question: Why is it that we do not need a larger space than the plane to solve Complex polynomial equations?
I have tried to find a sufficient answer through Google, but was not able. I also searched M.SE and could not find a sufficient answer. I am not a mathematician, so I am looking for an intuitive answer if possible.