  Quaternionic Clifford analysis is a recent new branch of Clifford analysis, a
higher dimensional function theory which refines harmonic analysis and
generalizes to higher dimension the theory of holomorphic functions in the
complex plane. So-called quaternionic monogenic functions satisfy a system of
first order linear differential equations expressed in terms of four
interrelated Dirac operators. The conceptual significance of quaternionic
Clifford analysis is unraveled by showing that quaternionic monogenicity can be
characterized by means of generalized gradients in the sense of Stein and
Weiss. At the same time, connections between quaternionic monogenic functions
and other branches of Clifford analysis, viz Hermitian monogenic and standard
or Euclidean monogenic functions are established as well.
