I think I have a complete list for all the commutative ones, maybe with possible repeats (I did try my best to make sure none are same up to isomorphism):
$\mathbb{R}^4 \simeq \begin{bmatrix}a&0&0&0\\0&b&0&0\\0&0&c&0\\0&0&0&d\end{bmatrix}$
$\mathbb{R}^2 \times \mathbb{C} \simeq \begin{bmatrix}a&0&0&0\\0&b&0&0\\0&0&c&-d\\0&0&d&c\end{bmatrix}$
$\mathbb{R}^2 \times \mathbb{R}[x]/(x^2) \simeq \begin{bmatrix}a&0&0&0\\0&b&0&0\\0&0&c&d\\0&0&0&c\end{bmatrix}$
$\mathbb{R} \times \mathbb{R}[x]/(x^3) \simeq \begin{bmatrix}a&0&0&0\\0&b&c&d\\0&0&b&c\\0&0&0&b\end{bmatrix}$
$\mathbb{C}^2 \simeq \begin{bmatrix}a&-b&0&0\\b&a&0&0\\0&0&c&-d\\0&0&d&c\end{bmatrix}$
$\mathbb{C} \times \mathbb{R}[x]/(x^2) \simeq \begin{bmatrix}a&-b&0&0\\b&a&0&0\\0&0&c&d\\0&0&0&c\end{bmatrix}$
$\mathbb{R}[x]/(x^4) \simeq \begin{bmatrix}a&b&c&d\\0&a&b&c\\0&0&a&b\\0&0&0&d\end{bmatrix}$
$\mathbb{R}[x,y]/(x^2,y^2+1) \simeq \mathbb{C}[x]/(x^2) \simeq \begin{bmatrix}a&-b&c&-d\\b&a&d&c\\0&0&a&-b\\0&0&b&a\end{bmatrix}$
and here are what I think are noncommutative ones:
$\mathbb{H} \simeq \mathbb{R}\langle x,y\rangle/(x^2+1,y^2+1,xy+yx)$
$\mathbb{R}\langle x,y\rangle/(x^2+1,y^2-1,xy+yx) \simeq \mathbb{R}\langle x,y\rangle/(x^2-1,y^2-1,xy+yx)$
$\mathbb{R}\langle x,y\rangle/(x^2-1,y^2,xy+yx)$
$\mathbb{R}\langle x,y\rangle/(x^2+1,y^2,xy+yx)$
$M_2(\mathbb{R})$
I am wondering is this is the complete list, I did try to make sure there are no repeats up to isomorphism, but there is no guarantee. And maybe the list isn't complete, maybe there is a subalgebra of $M_3(\mathbb{R})$ thats not on this list?