I have recently entertained the possibility of defining complex contour integration for the quaternions. I am somewhat aware that the Frobenius theorem dictates that no division algebra can exist in $\mathbb{R}^3$; however, Cayley-Dickinson constructions can generalize these division algebras to spaces of the form $\mathbb{R}^{2^n}$ at the expense of commutativity and later associativity. Even if the holomorphicity and analyticity are not equivalent (although they are for $\mathbb{C}$) generalizations of the Cauchy-Riemann equations have been made to higher dimensions with some success. [1] Additionally, a field analyzing differential quaternions exists in computer graphics, though I believe it is more concerned with aptly representing rotations in $\mathbb{R}^3$. [2] Would it be reasonable to believe that contour integration may be generalized to one of these division algebras by designing a hypercomplex function on a 1-manifold map (i.e., one that is diffeomorphic to a subset of $\mathbb{R}$.)
Although Liouville's theorem requires that the amount of holomorphic maps significantly decreases as the dimensionality increases and those that do exist must be representable as a composition of Mobius transforms, would a reasonable definition of quaternion complex integration be the sum of standard contour integration along each component of the one manifold for a given basis? (I also assume that the result would have to be invariant of the choice of basis in order to be well-defined.) In particular, I'm wondering if a generalization of contour integration could be used to define a biholomorphic map (if not a holomorphic map) in the quaternions or complex-like numbers spaces above? I'm very new to the topic and haven't the experience of many members on this website. As a semi-related question, why does the existence of quaternions not provide for the existence of division algebras in $\mathbb{R}^3$? I assume that taking the subset of all quaternions where only the same three components are nonzero would not suffice as the existence of a multiplicative inverse or closure under division would not hold.
As a final soft question, does anyone suggest textbooks for hypercomplex numbers? I had considered purchasing Hypercomplex Numbers by Isaiah Kantor. Are there any papers that expound on the notion of integration on Hypercomplex Manifolds described on Wikipedia? [3]
Thank you all.