Two vector spaces are said to be isomorphic iff there's an invertible linear map between them. It can be shown that isomorphic vectors spaces would have to have the same finite dimension or both be infinite dimensional. But what if they are over different fields? For example, would the trivial vector space over $\mathbb C$ be considered isomorphic to the trivial vector space over $\mathbb R$? Or would it not, since, if we let $T$ be the only possible linear map between them, $T(i0)\neq iT(0)=i0$, since $i0$ is not defined in the codomain (since $i$ is not a scalar in $\mathbb R$)?
Also, are there any other examples of vector spaces over different field that would be "isomorphic" like this?