As an $\mathbb{R}$-linear map, multiplication by complex number $z = a +bi$ can be represented by the matrix $$\begin{bmatrix} a && -b \\ b && a \end{bmatrix}$$ I do not know if there is a standard notation for these matrices as a subgroup of $GL_2(\mathbb{R})$. Let the subgroup of such matrices be denoted by $G$.
One way to think of complex analysis (perhaps not the best way) is as the study of solutions $f: \mathbb{R}^2 \to \mathbb{R}^2$ to the system of PDEs given by $ Df \in G$ (the Cauchy-Riemann equations). Clearly, there's much more because you have the algebra of complex numbers to play around with, but this is one aspect.
Complex Analysis is well known to be both do-able and very rich. Are there any analogs of results from complex analysis in other dimensions if we replace the group $G$ by some other subgroup of $GL_n(\mathbb{R})$? Even if there are not analogs, are there any interesting results? For instance, is there anything interesting about considering solutions $f: \mathbb{R}^n \to \mathbb{R}^n$ to $Df \in SO(n)$?
A "generalized analytic function" from an open subset $\Omega$ of $\mathbb{R}^n$ to $\mathbb{R}^n$ is a differentiable function $F :\Omega\rightarrow\mathbb{R}^n$ such that: $$\forall x\in \Omega, \left(\operatorname{div}(DF(x))=0\right)\land\left(\operatorname{curl}(DF(x))=0\right).$$ You can find a treatment of such a functions in the chapter 3 of the book Introduction to Fourier Analysis on Euclidean Spaces by Stein and Weiss. Further generalizations of this concept are also treated in chapter 6.