How to create a basis transformation matrix between "heterogenous" (...) bases? Consider the vector space $\mathbb{R}^2$ with bases $B_1 = \\{ e_1, e_2 \\}$, and $B_2 = \\{ e_1+e_2, e_1-e_2 \\}$. Because the vectors in $B_2$ are linear combinations of vectors in $B_1$, the transformation matrix $B_1 \rightarrow B_2$ $\begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\\ \frac{1}{2} & \- \frac{1}{2} \end{bmatrix}$ can be created. Now, consider the vector space $\mathbb{Q}(\sqrt{-1})$ with bases $B_1=\\{1, \sqrt{-1} \\}$, and $B2=\\{\begin{bmatrix} 1 & 0 \\\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\\ -1 & 0 \end{bmatrix} \\}$ then, how would I (formally) formulate the transformation matrix $B_1 \rightarrow B_2$ and $B_2 \rightarrow B_1$?

$B_2$ is not a subset of $\mathbb Q(i)$, so it is not a basis for the vector space $\mathbb Q(i)$. Now, maybe you are thinking of a certain isomorphism between a vector space of $2 \times 2$ matrices and $\mathbb Q(i)$, and maybe this isomorphism maps $B_2$ to the set $\tilde B_2 = \\{1, -i\\}$, which is in fact a basis for $\mathbb Q(i)$. It makes sense to ask what is the change of basis matrix from $B_1$ to $\tilde B_2$. And that is a question you probably know how to answer.