The quaternionic/hypercomplex-correspondence

Given a quaternionic manifold $M$ with a certain $\mathrm{U}(1)$-symmetry, we construct a hypercomplex manifold $M'$ of the same dimension. This construction generalizes the quaternionic Kahler/ hyper-Kahler-correspondence. As an example of this construction, we obtain a compact homogeneous hypercomplex manifold which does not admit any hyper-Kahler structure. Therefore our construction is a proper generalization of the quaternionic Kahler/hyper-Kahler-correspondence.