There is a visible sketch of a proof of the fundamental theorem of algebra. For the polynomial $f(z)=a_0+a_1z\cdots + a_nz^n$ over $\mathbb C$, with $a_0\neq 0$, define a function $g_r:[0,2\pi]\to\mathbb C$ by $g_r(\phi)=f(re^{i\phi})$. Given an open disk $\mathcal O_r\subset\mathbb C$ such that $0\notin\mathcal O_r\ni a_0$. If $r>0$ is small enough the image $\operatorname{Im} g_r\subset\mathcal O_r$. On the other hand, if $r$ is big enough the term $a_nz^n$ totally dominates and since the transformation of $\operatorname{Im} g_r$ as $r$ groves is continuous there must be an $r$ such that $0\in\operatorname{Im} g_r$.
Is it possible to make this proof rigorous with elementary methods and without an unreasonable effort?