We consider the Cauchy--Fueter complex of differential operators whose solution sheaf is the sheaf of regular functions of several quaternionic variables. We study a free resolution of this complex, we show the vanishing of some of its Ext-modules, and we obtain some new duality theorems which hint at a hyperfunction theory of several quaternionic variables. In particular, the vanishing of the first Ext-module, according to a classical idea of Ehrenpreis-Palamodov, gives a new proof that regular functions of several quaternionic variables cannot have compact singularities.