The monopole equations in the dual abelian theory of the N = 2 gauge-t
heory, recently proposed by Witten as a new tool to study topological
invariants, are shown to be the simplest elements in a class of instan
ton equations that follow from the improved topological twist mechanis
m introduced by the authors in previous papers. When applied to the N
= 2 sigma-model, this twisting procedure suggested the introduction of
the so-called hyperinstantons that are the solutions to an appropriat
e condition of triholomorphicity imposed on the maps q : M --> N from
a four-dimensional almost quaternionic world-manifold M to an almost q
uatemionic target manifold N. When gauging the sigma-model by coupling
it to the vector multiplet of a gauge group G, one gets instantonic c
onditions (named by us gauged hyperinstantons) that reduce to the Seib
erg-Witten equations for M = N = R(4) and G = U(1). The deformation of
the self-duality condition on the gauge-field strength due to the mon
opole-hyperinstanton is very similar to the deformation of the self-du
ality condition on the Riemann curvature previously observed by the au
thors when the hyperinstantons are coupled to topological gravity. In
this paper the general form of the hyperinstantonic equations coupled
to both gravity and gauge multiplets is presented.