We can encode the complex numbers as matrices by letting $1$ be the identity matrix and $i = \left( \begin{array}{clcr} 0 & -1\\1 & 0 \end{array} \right)$. From this fact, this means that complex analysis is equivalent to the analysis of matrices of the form $ \left( \begin{array}{clcr} a & -b\\b & a \end{array} \right)$.
If we removed the restriction on the form of the matrices, and instead studied all invertible $2\times2$ matrices, how much of the theorems from complex analysis would generalize to this new system? Is there any place where I could read more about this?
The set of real matrices of the form $\left(\begin{smallmatrix}a & -b \\ b & a\end{smallmatrix}\right)$ together with matrix addition and matrix multiplication form a field, that is, the field of complex numbers.
The set of all real $2\times 2$ matrices with the same addition and multiplication forms a unitary ring that is not commutative, does not admit inverses for all non-zero elements and hence misses a lot of properties that a field has.
Now the analysis for complex functions and such gadgets heavily depends on $\mathbb C$ being a field, so most of it will fail for functions over the ring of all real $2\times 2$ matrices.