  Given a K\"ahler manifold $M$ endowed with a Hamiltonian Killing vector field
$Z$, we construct a conical K\"ahler manifold $\hat{M}$ such that $M$ is
recovered as a K\"ahler quotient of $\hat{M}$. Similarly, given a
hyper-K\"ahler manifold $(M,g,J_1,J_2,J_3)$ endowed with a Killing vector field
$Z$, Hamiltonian with respect to the K\"ahler form of $J_1$ and satisfying
$\mathcal{L}_ZJ_2= -2J_3$, we construct a hyper-K\"ahler cone $\hat{M}$ such
that $M$ is a certain hyper-K\"ahler quotient of $\hat{M}$. In this way, we
recover a theorem by Haydys. Our work is motivated by the problem of relating
the supergravity c-map to the rigid c-map. We show that any hyper-K\"ahler
manifold in the image of the c-map admits a Killing vector field with the above
properties. Therefore, it gives rise to a hyper-K\"ahler cone, which in turn
defines a quaternionic K\"ahler manifold. Our results for the signature of the
metric and the sign of the scalar curvature are consistent with what we know
about the supergravity c-map.
