It is known that the structure of $p$-adic integers, $\Bbb{Z}_p$ is homeomorphic to the Cantor set, and $\Bbb{Q}_p$ is homeomorphic to the one-point deleted Cantor set (as I know, I don't certain it.) In particular, $\Bbb{Z}_p\cong\Bbb{Z}_q$ and $\Bbb{Q}_p\cong\Bbb{Q}_q$ even if $p$ and $q$ are distinct.
My question is:
Is $\Bbb{C}_p\cong\Bbb{C}_q$ even if $p$ and $q$ are distinct?
Since $\Bbb{C}_p$ is separable and a complete metric space by the metric given by $p$-adic norm, $\Bbb{C}_p$ is a Polish space. Are there well-known Polish spaces which are (topologically) homeomorphic to $\Bbb{C}_p$? (For example, $\Bbb{Z}_p$ is homeomorphic to the Cantor set, and Cantor set is a well-known Polish space.)
(If the tags of this question are inadequate, please tell me about it.)