  Quaternionic formulation of D=4 conformal group and of its associated
twistors and their relation to harmonic analyticity is presented.
Generalization of $SL(2,\cal{C})$ to the D=4 conformal group SO(5,1) and its
covering group $SL(2,\cal{Q})$ that generalizes the euclidean Lorentz group in
$R^4$ [namely $SO(3,1)\approx SL(2,\cal{C})$ which allow us to obtain the
projective twistor space $CP^3$] is shown. Quasi-conformal fields are
introduced in D=4 and Fueter mappings are shown to map self-dual sector onto
itself (and similarly for the anti-self-dual part). Differentiation of Fueter
series and various forms of differential operators are shown, establishing the
equivalence of Fueter analyticity with twistor and harmonic analyticity. A
brief discussion of possible octonion analyticity is provided.
