Why is it that Complex Numbers are algebraically closed? 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.
Of course there are many proofs, and perhaps some others will post the most attractive proofs,  but I think you are looking for an intuitive explanation that would somehow make the result seem less surprising. 

One such explanation, I think, is the simple observation that reals already go a long way towards being algebraically closed---they are a real closed field---since every odd-degree polynomial over $\mathbb{R}$ has a root in $\mathbb{R}$. This follows immediately from the intermediate value theorem, since in the large scale every odd degree polynomial moves from $-\infty$ to $\infty$ or conversely and hence must cross the axis.
One definition of the complex numbers is that they are the algebraic closure of the reals.

In other words, start with the reals and write down some degree-$d$ polynomial that has fewer than $d$ roots (incl multiplicities).  Call one of these roots $x$.  Now extend the reals with $x$.  After you continue this (infinite) process, you have $\mathbb{C}$.

If you're willing to buy this as the definition of $\mathbb{C}$, then it's trivial to see that it's algebraically closed.