The matrix $\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right)$ has no square root.
Polynomials make sense for continuous complex functions on a space. If that space is $\mathbb R$, then polynomial equations with complex coefficients are solvable. If that space is $\mathbb C$ or $S^1$ then $g^2 = f$ may not be solvable.
