Introduction

Abstract

The best known extension of the field of complex numbers to the four-dimensional setting is the skew field of quaternions, introduced by W.R. Hamilton in 1844, [36], [37]. Quaternions arise by considering three imaginary units, i, j, k that anticommute and such that ij = k. The beauty of the theory of quaternions is that they form a field, where all the customary operations can be accomplished. Their blemish, if one can use this word, is the loss of commutativity. While from a purely algebraic point of view, the lack of commutativity is not such a terrible problem, it does create many difficulties when one tries to extend to quaternions the fecund theory of holomorphic functions of one complex variable. Within this context, one should at least point out that several successful theories exist for holomorphicity in the quaternionic setting. Among those the notion of Fueter regularity (see for example Fueter’s own work [27], or [97] for a modern treatment), and the theory of slice regular functions, originally introduced in [30], and fully developed in [31]. References [97] and [31] contain various quaternionic analogues of the bicomplex results presented in this book.