Simple Proof of FT of Algebra

Question

Which proof of the Fundamental Theorem of Algebra requires minimum mathematical maturity and has the best chances to be understood by an amateur with knowledge of complex numbers and polynomials?

Answer 1

Suppose the polynomial has no zeros. Then every value of $f(z)$ has a direction.
Draw a large circle, where $z^n$ dominates over the other terms in the polynomial. As you travel around the circle, the direction of $f(z)$ is near the direction of $z^n$, so it rotates $n$ times as $z$ goes once around the circle.
Now gradually shrink the circle, until it is nearly nothing. $f(z)$ is nearly constant as you go around the circle, so it doesn't rotate at all as you go around the circle.
The number of rotations must change from $n$ to $0$ at some point. But unless $f(z)=0$ somewhere, the number of rotations is continuous. So $f(z)=0$ somewhere.