If the elegant Weierstrass approximation theorem true for complex functions? Meaning, if $\displaystyle f(z)$ be a continous function on a compact set $\displaystyle S$ (non-empty) then for any $\displaystyle \epsilon > 0$ there exists a polynomial $\displaystyle g(z)$ such that $\displaystyle |f(z)-g(z)| < \epsilon$.