  We define a very general notion of regularity for functions taking values in
an alternative real $*$-algebra. Over Clifford numbers, this notion subsumes
the well-established notions of monogenic function and slice-monogenic
function. Over quaternions, in addition to subsuming the notions of
Fueter-regular function and of slice-regular function, it gives rise to an
entirely new theory, which we develop in some detail.
