  We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose
quaternionic structure is trivialized by a hypercomplex structure, a
hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally
symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to
the quaternionic projective space or to the quaternionic hyperbolic space. We
describe locally the hyper-Hermitian quaternionic Kaehler manifolds with closed
Lee form and show that the only complete simply connected such manifold is the
quaternionic hyperbolic space.
