Quarternionic Analysis What is/are the current understanding/opinions about Quarternionic Analysis as a generalization of Complex Analysis with respect to a "Quarternionic Residue Calculus" (if such a thing exists)? i.e. does a "quarternionic residue calculus" exist, and if not what are the opinions that this might be useful, if indeed possible?
For example, residue calculus is great if we can find a "nice" contour to work with, but sometimes this does not seem possible. I'm just wondering if a Quarternionic Analysis would allow for 4-dimensional "contours" (i.e. surfaces) in the "Quarternion-plane" (i.e. 4-dimensional space) that might allow the evaluation of definite integrals which seem otherwise unsolvable by the residue calculus of complex analysis.
 A: A while back, when asking myself the same question, I found this website to be very informative. The wiki article was also useful, if only for its bibilography.
You might also have better luck if you search for "hypercomplex analysis." In googlebooks I found Generalized Analytic Automorphic Forms in Hypercomplex Spaces By Rolf S. Krausshar. On page 21 it states Theorem 1.23 (Residue theorem) for certain finite dimensional differentiable manifolds, and on page 22 it says:
Also the residue theorem can be generalized to the more general context dealing with manifolds of singularities. See [61,63] for the quaternionic case...
I would give you the two references but I can't see the bibliography. If I were in your position I'd definitely find this book :)
I did not get very deep into the subject: I just satisfied my curiosity about whether or not people have been thinking about analyticity, differentiation, integration and all other things from real and complex analysis in the context of the quaternions, and it appears that they have. 
