In particular I wondered about the following: The Weierstrass-function $\mathcal{W}$ is continuous and nowhere differentiable. By the Stone-Weierstrass-Theorem we can approximate $\mathcal{W}$ on $[0,1]$ uniformly by real polynomials. Let $p_n(x)$ be such a sequence of polynomials. Now we consider the $p_n$ as complex polynomials. On $[0,1]$ the $p_n$ of course still converge pointwise to $\mathcal{W}$. On $[0,1]\times i[-\frac{1}{2},\frac{1}{2}]\subseteq\mathbb{C}$ however this convergence can not be uniform anymore, as this would imply holomorphy on $(0,1)\times i(-\frac{1}{2},\frac{1}{2})$ which would imply real differentiability on $(0,1)$.
I find this very unintuitive, so i would like to see a concrete example of a sequence of polynomials converging uniformly on some $[a,b]\subseteq \mathbb{R}$ but not converging uniformly on any $[a,b]\times i[-\epsilon,\epsilon]\subseteq\mathbb{C}$, if possible with a direct verification that this is (not) the case.