In this post the OP wonders how one can prove the FTA using the hairy ball theorem, after having read a book of Ian Stewart containing an allusion to such a proof. The answer I gave could be added to the current list of proofs as well. There is no doubt that this proof is known. However, it does not seem to be mentioned in the other items of this list.

The main idea of proof is already present when one wants to show that a polynomial $f\in\mathbf C[z]$ of degree $2$ has a zero using the hairy ball theorem. Suppose that it does not have any zeros. Then the holomorphic vector field $$ f\tfrac{\partial}{\partial z} $$ is a nonvanishing vector field on the Riemann sphere $S^2$. This is because $f$ has a pole of order $2$ at infinity, while $\partial/\partial z$ has a zero of order $2$ at infinity. The contradiction comes from the hairy ball theorem that states that such vector fields cannot exist. The general case of a polynomial $f$ of even degree $2d$ can be proved by arguing that the vector field $$ \sqrt[d]{f\cdot\left(\tfrac{\partial}{\partial z}\right)^{\otimes d}} $$ is well defined, and nonvanishing on $S^2$ if $f$ does not have any zeros in $\mathbf C$. See here for more details.

In this post the OP wonders how one can prove the FTA using the hairy ball theorem, after having read a book of Ian Stewart containing an allusion to such a proof. The answer I gave could be added to the current list of proofs as well. There is no doubt that this proof is known. However, it does not seem to be mentioned in the other items of this list.

The main idea of proof is already present when one wants to show that a polynomial $f\in\mathbf C[z]$ of degree $2$ has a zero using the hairy ball theorem. Suppose that it does not have any zeros. Then the holomorphic vector field $$ f\tfrac{\partial}{\partial z} $$ is a nonvanishing vector field on the Riemann sphere $S^2$. This is because $f$ has a pole of order $2$ at infinity, while $\partial/\partial z$ has a zero of order $2$ at infinity. The contradiction comes from the hairy ball theorem that states that such vector fields cannot exist. The general case of a polynomial $f$ of even degree $2d$ can be proved by arguing that the vector field $$ \sqrt[d]{f\cdot\left(\tfrac{\partial}{\partial z}\right)^{\otimes d}} $$ is well defined, and nonvanishing on $S^2$ if $f$ does not have any zeros in $\mathbf C$. See here for more details.