Is the complex plane homeomorphic to $\mathbb{R}^2$? Is the set of complex numbers homeomorphic to $\mathbb{R}^2$? 
They are isomorphic. Are they homeomorphic? 

Yes. The topologies of both spaces are induced by the Pythagorean norm, so the bijection
$$
\left\{
\begin{matrix}
\Bbb{R}^{2} & \to     & \Bbb{C} \\
(x,y)       & \mapsto & x + i y
\end{matrix}
\right\}
$$
maps open balls of $ \Bbb{R}^{2} $ to open balls of $ \Bbb{C} $ and vice-versa.
