  We analyse the geometry of four-dimensional bosonic manifolds arising within
the context of $N=4, D=1$ supersymmetry. We demonstrate that both cases of
general hyper-K\"ahler manifolds, i.e. those with translation or rotational
isometries, may be supersymmetrized in the same way. We start from a generic
N=4 supersymmetric three-dimensional action and perform dualization of the
coupling constant, initially present in the action. As a result, we end up with
explicit component actions for $N=4, D=1$ nonlinear sigma-models with
hyper-K\"ahler geometry (with both types of isometries) in the target space. In
the case of hyper-K\"ahler geometry with translational isometry we find that
the action possesses an additional hidden N=4 supersymmetry, and therefore it
is N=8 supersymmetric one.
