Recent Developments for Regular Functions of a Hypercomplex Variable

In this paper we survey a series of recent developments in the theory of functions of a hypercomplex variable. The central idea underlying these developments consists in requiring a function to be holomorphic on suitable slices of the space on which the function itself is defined. Specifically, we apply this approach to functions defined on the space H of quaternions, on the space O of octonions, and finally on the Clifford algebra of type (0,3), denoted Cl(0, 3). The properties of these functions resemble those of holomorphic functions, and yet the different nature of the three algebras on which we work introduces new and exciting phenomena.