In the recent paper
Anton, R.; Mihalache, N.; Vigneron, F., A Short ODE Proof of the Fundamental Theorem of Algebra., Math. Intelligencer (2023).
the authors give a proof based on the analysis of the attractive points of the initial value problem:
$$\varphi_z(0)=z, \quad \varphi_z^{\prime}(t)=-\frac{P\left(\varphi_z(t)\right)}{P^{\prime}\left(\varphi_z(t)\right)},$$
which is a continuous-time version of Newton's method for computing the zeros of the polynomial $P$.