  As a generalization of Riemannian submersions, horizontally conformal
submersions, semi-invariant submersions, h-semi-invariant submersions, almost
h-semi-invariant submersions, conformal semi-invariant submersions, we
introduce h-conformal semi-invariant submersions and almost h-conformal
semi-invariant submersions from almost quaternionic Hermitian manifolds onto
Riemannian manifolds.
  We study their properties: the geometry of foliations, the conditions for
total manifolds to be locally product manifolds, the conditions for such maps
to be totally geodesic, etc. Finally, we give some examples of such maps.
