In Clifford analysis, massless field equations for higher spin fields were studied from various points of view. In particular, the classical and the most interesting case consists of such equations in dimension four. They are elliptic versions of massless field equations on Minkowski space used systematically for a long time in theoretical physics. The most important class of such equations have a property that their solutions are componentwise harmonic. The classification of such equations is going back to the work of Lars G\aa rding in the middle of the last century. The most important cases are those of spin 1/2 fields (which is the case of the classical Clifford analysis) and spin 1 fields, which are very closely related to the Maxwell equations. The Fischer decomposition for spin 1 fields was described recently with its complete structure of invariant operators. The aim of this contribution is to discuss the Fischer decomposition for spin 3/2 fields in the framework of Clifford analysis and to complement a completely different approach coming to these equation from point of veiw of the Howe duality for Lie superalgebras.