The aim of this thesis is to combine the strengths of two approaches: merging the algebraic structure of the interplay between the variable and the differential operator in Clifford analysis with the rich kernel of the differential operator in slice Clifford analysis. To put it short: Clifford analysis will be performed in a slice framework.

When talking about an algebraic structure, we are aiming for one precise structure in particular, namely that of osp(1|2). This Lie superalgebra would allow to find a representation theoretic interpretation of various function space decompositions (see [32]). Moreover it would pave the way for the introduction of a generalised Fourier (see e.g. the review [30]) and Segal-Bargmann transform to this framework.