Is a scalar extension of an $(t)$-adically complete module complete?

Question

Let $M$ be a $(t)$-adically complete $\mathbb C[[t]]$-module, say $M$ is a topologically free $\mathbb C[[t]]$-module. Let $\mathbb C((t))$ be the field of formal Laurent series. Is then the scalar extension $$M\otimes_{\mathbb C[[t]]}\mathbb C((t))$$ $(t)$-adically complete? It is hard for me to verify this since I cannot figure out if the inverse limit commutes in this case with the tensor product. Does it make sense at all to talk about $(t)$-adic completeness in the case of a $\mathbb C((t))$-vector space?