Clifford analysis versus its quaternionic counterparts

For a positive integer n let Cl₀,_n be the universal Clifford algebra with the signature (0,n). The name Clifford analysis is usually referred to the function theories for functions in the kernels of the two operators: the (Cliffordian) Cauchy- Riemann operator and the Dirac operator. For n = 2, Cl₀,₂ becomes the skew-field of Hamilton's quaternions for which the two operators are widely known: the Moisil-Théodoresco and the Fueter operators. We establish the precise relations between the Moisil-Théodoresco operator and the Dirac operator for Cl₀,₃. It turns out that the case of the Cauchy- Riemann operator for Cl₀,₃ and the Fueter operator is more sophisticated, and we describe the peculiarities emerging here.