The extension of the Cauchy-Riemann equations to functions of a quaternionic variable is known as the $\textbf{Cauchy-Riemann-Fueter equations}$, however note that the concept of differentiability and derivative on the quaternions is defined in a radically different way due to the following problem:
If we try to define the (left/right) derivative (their non-commutativity forces us to define a left and a right version of the derivative) of a quaternionic function as:
$$\lim_{h \to 0} h^{-1}(f(q_0 + h) - f(q_0))$$
and
$$\lim_{h \to 0} (f(q_0 + h) - f(q_0))h^{-1}$$
respectively, then we will see that the only functions for which these limits will exist will be left and right (respectively) linear functions, i.e functions of the form $f(q) = a + qb$ and $f(q) = a + bq$. This fact was proved originally by V.C.A Ferraro at the beginning of the 20th century, but a more modern proof can be found in the famous paper by Sudbery.
This actually introduces a problem: there are several equally valid formulations of the concept of a differentiable function of a quaternionic variable; I covered 4 approaches in my monography, but here I will just give the one due to Fueter (however I will define it in the way in which it is done in Sudbery's paper).
A quaternionic function $f:\mathbb{H} \to \mathbb{H}$ is said to be left-regular in $q \in \mathbb{H}$ if there exists a quaternion $f_l'(q)$ that we will call $\textbf{the left derivative of f in q}$ such that:
$$d(dq \wedge dq f) = Dq f_l'(q)$$
where here $Dq$ is a quaternionic differential form defined as
$$Dq := dx \wedge dy \wedge dz - \frac{1}{2}\epsilon_{ijk}e_i dt \wedge dx_j \wedge dx_k$$
Where here $e_i$ is the ith quaternion imaginary unit (i.e $e_1=i, e_2 = j$ and so on) and $\epsilon_{ijk}$ is the levi civita symbol. Furthermore I employed Einstein's summation convention to make the expression simpler and more compact.
Understanding what this equation means intuitively requires a good knowledge of the genetic development of hypercomplex analysis and quaternionic analysis and here I don't have the time to explain everything in depth; for this you can check my monography, that I also linked above. In part it boils down to the important geometric properties of the Dq differential form, and the proofs of these are provided in Sudbery's paper.
For these classes of functions, we can then prove the (left/right) $\textbf{Cauchy-Riemann-Fueter equations}$, which state the following:
- Left Cauchy-Riemann-Fueter equation (also commonly known as left CRF): a quaternionic function is left-regular in $q \in \mathbb{H}$ if and only if, in $q$ we have:
$$\frac{\partial f}{\partial t} + i\frac{\partial f}{\partial x} + j\frac{\partial f}{\partial y} + k\frac{\partial f}{\partial z} = 0 $$
- Right Cauchy-Riemann-Fueter equation (also commonly known as right CRF): a quaternionic function is right-regular in $q \in \mathbb{H}$ if and only if, in $q$ we have:
$$\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}i + \frac{\partial f}{\partial y}j + \frac{\partial f}{\partial z}k = 0 $$
The proof of these 2 theorems is pretty simple and can be found here (page 217).
This gives rise to two theories, dual to each other, of left and right Fueter-regular functions, for which many extensions of famous results of complex analysis will hold (such as Cauchy's theorem, Cauchy's integral formula, Morera's theorem, Liouville's theorem, Laurent and Taylor series expansions and more).
Cauchy's theorem assumes the following form (for left regular functions):
$$\int_C Dq f(q) = 0 $$
Where here $f: U \to \mathbb{H}$ is a left regular quaternionic function, $C$ is a smooth 3-chain homologous to 0 in the singular differentiable homology of the quaternionic domain $U$ and $Dq$ is the form from before.
Laurent series are of the following form:
$$f(q) = \sum_{n=0}^\infty \sum_{\nu \in \sigma_n} [P_\nu(q-q_0)a_\nu + G_\nu(q-q_0)b_\nu]$$
where the coefficients are given by integral expressions similar to those of complex laurent series, $G_\nu$ is the $\nu$-th mixed-differential of the quaternionic equivalent of the cauchy kernel, and $P_\nu$ are what is known as the "polynomial base" of the module $U_n$, the module of regular homogeneous functions of degree $n$.
The functions involved in the expansion are not, strictly speaking, polynomials and Laurent polynomials like in the complex case, but as I tried to highlight before they mirror many of some properties polynomials and Laurent polynomials have in $\mathbb{C}$.
All these theorems require preliminary concepts that in turn require others, and I'm aware that without introducing these concepts first the notions that I'm giving could feel meaningless.
I know this question is quite old, but I hope this answer was helpful and that I was able to help you and your curiosity in some way. The amount of information I can include inside a stackexchange answer is very limited, and for this reason I cited numerous resources you can check to delve more deeply into the subject.
