  This is a continuation of a series of papers [FL1, FL2, FL3], where we
develop quaternionic analysis from the point of view of representation theory
of the conformal Lie group and its Lie algebra. In this paper we continue to
study the quaternionic analogues of Cauchy's formula for the second order pole.
These quaternionic analogues are closely related to regularization of
infinities of vacuum polarization diagrams in four-dimensional quantum field
theory. In order to add some flexibility, especially when dealing with Cauchy's
formula for the second order pole, we introduce a one-parameter deformation of
quaternionic analysis. This deformation of quaternions preserves conformal
invariance and has a geometric realization as anti de Sitter space sitting
inside the five-dimensional Euclidean space. We show that many results of
quaternionic analysis - including the Cauchy-Fueter formula - admit a simple
and canonical deformation. We conclude this paper with a deformation of the
quaternionic analogues of Cauchy's formula for the second order pole.
