Consider $k= \mathbb{F}_q(t)$ and let $p$ be an irreducible polynomial in $\mathbb{F}_q[t]$. Consider the completions of $k$ with respect to $p$ and the $\infty$-adic completion, with notation $k_p$ and $k_{\infty}$.
Are the algebraic closures of these two completions isomorphic? My reasoning is that they should be: clearly $k_{\infty}$ is isomorphic to $\mathbb{F}_q((t))$ and $k_p$ has to be a finite extension of $\mathbb{F}_q((t))$ since it is a local field of characteristic $p$. Hence an algebraic closure of $k_p$ is also an algebraic closure of $k_{\infty}$.
Is this reasoning correct? If it is, is it an isomorphism as topological fields?