The standard definition of Linear Transformation says:
It is easy to observe that the conditions hold even if the field of $U$ is super field of $V$'s field. For example $V$ as a space over the reals and $U$ as a space over complex numbers should work as stated here in a comment.
So my question is :
What exactly fails if we take $U$'s field to be the superfield of $V$'s field in stead of the usual definition which takes the same field?
If nothing fails can we safely modify the definition without effecting the other related concepts?
If $U$ is a superfield of $V$, then every $U$ vector space is also a $V$ vector space. And thus we have linear transformations (more exactly: $V$-linear transformations) from $V$-vector spaces to $U$-vector spaces.
However, when considering linear transformations it is important to consider the vector spaces over the same field because important properties change if the base field is different.
For example, $\mathbb R^3$ is finite dimensional as $\mathbb R$-vector space, but infinite-dimensional as $\mathbb Q$-vector space. Therefore e.g. a linear transformation $\mathbb Q^3\to\mathbb R^3$ cannot be surjective, as one might naively think if just looking at the dimension without considering the base field ($\mathbb Q^3$ has the same dimension as $\mathbb Q$-vector space as $\mathbb R^3$ has as $\mathbb R$-vector space).
Another example, $\mathbb R$ and $\mathbb R^2$ are isomorphic as $\mathbb Q$-vector spaces, but not as $\mathbb R$-vector spaces.