TITLE: riemann mapping theorem for skew-fields of quaternions and beyond
QUESTION [7 upvotes]: Is there any known generalization of the riemann mapping theorem over skew-fields of quaternions and beyond or at least a conjectured formulation of it?
In a less focused way, how far does the main results of complex analysis and fourier analysis carry over to quaternions.

REPLY [5 votes]: Let $\mathbb{H}$ denote the field of quaternions and $\mathbb{S}$ denote the unit sphere of imaginary quaternions. We have a Riemann mapping theorem for axially symmetric sets inside the field of quaternions $\mathbb{H}$. The set $\Omega$ is axially symmetric if the set $\{ x+Jy~|~J\in\mathbb{S} \},~J\in\mathbb{S}$ are contained in $\Omega$ (we have rotation invariance around the axis). Let $\Omega$ be an axially symmetric open set in $\mathbb{H}$. Extend $\Omega$ to $\mathbb{C}_I$ (the restricted complex plane that can be identified with $\mathbb{C}$) for $I\in\mathbb{S}$. Then if $\mathbb{B}$ is the open unit ball, then we have a bijective mapping of a quaternionic slice function $f$ that bijectively maps from $\Omega$ to $\mathbb{B}$. 
Check out the following paper by Sorin Gal and Irene Sabadini:
http://arxiv.org/pdf/1406.5516.pdf