A continuous bijection between two complete metric spaces that is not a homeomorphism.

The identity map from $\mathbb R$ with discrete metric into $\mathbb R$ with usual metric is a continuous bijection which is not a homeomorphism. Both spaces are complete.

First, observe that $[0,1)$ is homeomorphic to $[0,\infty)$ and clearly $[0,\infty)$ is complete because is a closed subset of a complete metric space ($\mathbb{R}$). Take $f:[0,\infty)\to[0,1)$ such homeomorphism. Consider the function $g:[0,1)\to\mathbb{S}^1$ defined by $g(x)=(\cos(2\pi x),\sin(2\pi x))$. It's not to hard to prove that $g$ is continuous and biyective. Thus, $g\circ f:[0,\infty)\to\mathbb{S}^1$ is continuous and biyective function and the domain is complete but $g\circ f$ is not an homeomorrphism because $\mathbb{S}^1$ is compact and $[0,\infty)$ isn't.