I'll approach this question from the wider context of finding bases for vector spaces of $4\times4$ squares of various kinds of magic.
First, following user1551, let's have
$\mathbf{b}_0=\begin{bmatrix}
1&1&1&1\\1&1&1&1\\1&1&1&1\\1&1&1&1
\end{bmatrix}$
so that we can easily shift a magic square up or down to make its lowest element what we want. (Or to put a desired value in the top left corner; all my other basis vectors have 0 there.)
Next, I add Grogono's four carpets which are collectively a basis for the 4-dimensional vector space of anchored compact complete $4\times4$ magic squares. (Anchored means that the top left element is 0.) With $\mathbf{b}_0$, we have a basis for the 5-dimensional vector space of compact complete ones (including unanchored).
$\mathbf{b}_1=\begin{bmatrix}
0&0&1&1\\1&1&0&0\\0&0&1&1\\1&1&0&0
\end{bmatrix}$,
$\mathbf{b}_2=\begin{bmatrix}
0&1&0&1\\0&1&0&1\\1&0&1&0\\1&0&1&0
\end{bmatrix}$,
$\mathbf{b}_3=\begin{bmatrix}
0&1&1&0\\1&0&0&1\\0&1&1&0\\1&0&0&1
\end{bmatrix}$,
$\mathbf{b}_4=\begin{bmatrix}
0&1&0&1\\1&0&1&0\\1&0&1&0\\0&1&0&1
\end{bmatrix}$.
With most of my later basis vectors, the point of adding the vector to the basis is to extend the vector space to include squares which lack a certain magic property possessed by every square which is in the vector space so far. These magic properties are illustrated below. Let
$D=\begin{bmatrix}
a&b&c&d\\b&a&d&c\\c&d&a&b\\d&c&b&a
\end{bmatrix}$,
$Q=\begin{bmatrix}
a&a&b&b\\a&a&b&b\\d&d&c&c\\d&d&c&c
\end{bmatrix}$,
$M=\begin{bmatrix}
a&b&b&a\\d&c&c&d\\d&c&c&d\\a&b&b&a
\end{bmatrix}$,
$T=\begin{bmatrix}
a&b&a&b\\d&c&d&c\\a&b&a&b\\d&c&d&c
\end{bmatrix}$.
Say a $4\times4$ square $S$ is '$D$-magic' if, for each of the symbols $a, b, c$ and $d$, the sum of $S$'s entries in the four locations containing that symbol in $D$ is $S$'s magic sum. And similarly for $Q, M$ and $T$. And say a $4\times4$ square is '$R$-magic' if each row is magic, and '$C$-magic' if each column is magic.
Then $RCD$-magic squares are magic squares with the additional property that broken diagonals that break 2 and 2 are magic. The vector space of them has 7 dimensions: 2 more. The following two squares will serve to extend the basis:
$\mathbf{b}_5=\begin{bmatrix}
0&0&1&1\\1&1&0&0\\1&1&0&0\\0&0&1&1
\end{bmatrix}$,
$\mathbf{b}_6=\begin{bmatrix}
0&1&1&0\\0&1&1&0\\1&0&0&1\\1&0&0&1
\end{bmatrix}$.
Next comes the OP's vector space: $4\times4$ magic squares. This has 8 dimensions: 1 more.
$\mathbf{m}=\begin{bmatrix}
0&0&1&1\\0&1&0&1\\1&0&1&0\\1&1&0&0
\end{bmatrix}$
can extend the basis. Incidentally, every $4\times4$ magic square is $M$-magic.
Return to the 7-dimensional vector space of $4\times4$ $RCD$-magic squares. $RCD$-magic squares are also $QMT$-magic. If $DT$-magic is dropped, the vector space now has 8 dimensions; we may extend the basis with
$\mathbf{b}_7=\begin{bmatrix}
0&1&0&1\\1&0&1&0\\0&1&0&1\\1&0&1&0
\end{bmatrix}$. If $Q$-magic is dropped, the vector space now has 9 dimensions; we may extend the basis with
$\mathbf{b}_8=\begin{bmatrix}
0&0&1&1\\0&0&1&1\\1&1&0&0\\1&1&0&0
\end{bmatrix}$. If $M$-magic is dropped, the vector space now has 10 dimensions; we may extend the basis with
$\mathbf{b}_7=\begin{bmatrix}
0&1&1&0\\1&0&0&1\\1&0&0&1\\0&1&1&0
\end{bmatrix}$.
To summarise,
compact $4\times4$ magic squares are the 5-dimensional vector space $\langle \mathbf{b}_0,\dots,\mathbf{b}_4\rangle$
$4\times4$ $RCD$-magic squares are also $QMT$-magic, and are the 7-dimensional vector space $\langle \mathbf{b}_0,\dots,\mathbf{b}_6\rangle$
$4\times4$ magic squares are the 8-dimensional vector space $\langle \mathbf{b}_0,\dots,\mathbf{b}_6, \mathbf{m}\rangle$
$4\times4$ $RCQM$-magic squares are the 8-dimensional vector space $\langle \mathbf{b}_0,\dots,\mathbf{b}_7\rangle$
$4\times4$ $RCM$-magic squares are the 9-dimensional vector space $\langle \mathbf{b}_0,\dots,\mathbf{b}_8\rangle$
$4\times4$ $RC$-magic squares (i.e. semi-magic squares) are the 10-dimensional vector space $\langle \mathbf{b}_0,\dots,\mathbf{b}_9\rangle$
