Having just learned the definition of analytic functions, I found it surprising and somewhat counterintuitive that the set of functions over the complex plane whose real and imaginary parts each satisfy Laplace's Equation $\nabla^{2} = 0$ should have such incredible significance, since this is basically a set of two harmonic functions over $\mathbb{R}$. Does this extend to higher dimensions of hypercomplex numbers such as quaternions?
More formally, is there some generalization of the space of analytic functions to higher dimensions $ 2^n | n \in \mathbb{N}, n > 1$ where each component function individually satisfies Laplace's Equation? Do these functional spaces yield similar properties to that enumerated in the field of complex analysis, or are there any salient differences?
In the book Introduction to Fourier Analysis on Euclidean Spaces, by E.M. Stein and G. Weiss, a "generalized analytic function" is defined as a function $u=(u_1,\dots,u_n)\colon D\subset\mathbb{R}^n\to\mathbb{R}^n$ such that $$ \sum_{i=1}^n\frac{\partial u_i}{\partial x_i}=0,\quad \frac{\partial u_i}{\partial x_j}=\frac{\partial u_j}{\partial x_i},\quad1\le i,j\le n. $$ This is equivalent to $$ \operatorname{div}u=0,\quad \operatorname{curl}u=0. $$