It is possible to define homomorphisms between vector spaces with different fields? I understand that an homomorphism between vector spaces must preserve the sum between vectors and the scalar multiplication.
By example, let vector spaces $(E,\Bbb Q)$ and $(F,\Bbb Q[\sqrt 2])$, then we can define something like
$$f:E\to Q$$
such that $f(\lambda v+\mu w)=\lambda\sqrt2 f(v)+\mu\sqrt2 f(w)$ for $\lambda,\mu\in\Bbb Q$ and $\lambda\sqrt 2,\mu\sqrt2\in\Bbb Q[\sqrt 2]$. This would be a homomorphism between vector spaces but not a linear map, right?
Or if we take some vector spaces $(A,\Bbb R)$ and $(B,\Bbb C)$ and we set a function
$$g:A\to B$$
such that $g(r a+s b)=r g(a)+sf(b)$ for $a,b\in A$ and $r,s\in \Bbb R$ this preserve the operations of vector spaces too because $\Bbb R\subset\Bbb C$.
These examples would be correctly called as homomorphisms between vector spaces (with no necessarily the same field)?
 A: By definition, linear maps of vector spaces can only exist between vector spaces over the same field. You could define a new class of maps between pairs of the form $(V,K)$ where $V$ is a vector space over $K$ with the property you mention (or more precisely, you would probably want pairs of maps $(f,g) : (V,K)\to(W,L)$, where $f : V\to W$ is a homomorphism of abelian groups, $g : K\to L$ is a homomorphism of rings and $f(av + bw) = g(a)f(v) + g(b)f(w)$ for $a,b\in K$, $v,w\in V$), but they would no longer be linear maps of vector spaces (although the data of such a map $(f,g) : (V,K)\to (W,L)$ would be equivalent to the data of a linear map $V\to W$ when $K = L$ and the map $g : K\to L$ is the identity).
There are other sorts of ways you might try to make this work as well: if $M$ is an $R$-module and $N$ is an $S$-module, and you have a morphism of rings $R\to S$, you can give $N$ the structure of an $R$ module via the homomorphism, and then you could talk about a morphism $M\to N$ of $R$-modules. In the world of vector spaces, this would be the same as starting with a vector space $V/K$ and a vector space $W/L$, where $L$ is an extension of $K$, and then via restriction, considering $W$ as a vector space over $K$, and looking at maps $V\to W$ where $W$ is considered as a $K$-vector space.
A: Homomorphisms between vector spaces $V$, $W$ (over the same field $k$) are by definition the linear maps between those vector spaces. Linearity itself, is also frequently mentioned with respect to the underlying field i.e. we say "$k$-linear map". In the level of the Category theory, these are the morphisms in the category of $k$-vector spaces. 
The maps you are speaking about in your post, are neither homomorphisms nor linear. In fact, they may face various problems in the following sense: Let such a map $g:A\to B$, between $(A,\Bbb R)$ and $(B,\Bbb C)$ and let it be bijective. Then it possesses an inverse map $g^{-1}$ (in the set-theoretic sense). Would this inverse, be "linear" in the sense defined in your post? It would actually not even be well-defined. Let alone the case in which the underlying fields might be completely different (as sets). 
