  Charged BPS hypermultiplets can develop a non-trivial self-interaction in the
Coulomb branch of an N=2 supersymmetric gauge theory, whereas neutral BPS
hypermultiplets in the Higgs branch may also have a non-trivial
self-interaction in the presence of Fayet-Iliopoulos terms. The exact
hypermultiplet low-energy effective action (LEEA) takes the form of the
non-linear sigma-model (NLSM) with a hyper-K"ahler metric. A non-trivial scalar
potential is also quantum-mechanically generated at non-vanishing central
charges, either perturbatively (Coulomb branch), or non-perturbatively (Higgs
branch). We calculate the effective scalar potentials for (i) a single charged
hypermultiplet in the Coulomb branch and (ii) a single neutral hypermultiplet
in the Higgs branch. The first case corresponds to the NLSM with the Taub-NUT
(or KK-monopole) metric for the kinetic LEEA, whereas the second one is
attached to the NLSM having the Eguchi-Hanson instanton metric.
