There are several analogues of the theory of one complex variable, when the values of the functions are taken in the division algebra H of quaternions, or in a suitable Clifford algebra. These theories rely on the construction of operators which somehow imitate the Cauchy-Riemann operator; in the quaternionic case one uses the Cauchy-Fueter operator, and in the Clifford case one uses the Dirac operator. The extension to several variables has remained elusive for a long time, but it can in fact be achieved if one considers these systems from the point of view of their algebraic properties. The analysis of such operators from the point of view of the Palamodov-Ehrenpreis Fundamental Principle allows the construction of a non-trivial theory in several variables. This talk will discuss the strength of this approach, as well as some of the questions which remain open, and will be concluded with a new twist on these theories.