  We use the manifestly N=2 supersymmetric, off-shell, harmonic (or twistor)
superspace approach to solve the constraints implied by four-dimensional N=2
superconformal symmetry on the N=2 non-linear sigma-model target space, known
as the special hyper-K"ahler geometry. Our general solution is formulated in
terms of a homogeneous (of degree two) function of unconstrained (analytic)
Fayet-Sohnius hypermultiplet superfields. We also derive the improved (N=2
superconformal) actions for the off-shell (constrained) N=2 projective
hypermultiplets, and relate them (via non-conformal deformations) to the
asymptotically locally-flat (ALF) A_k and D_k series of the gravitational
instantons. The same metrics describe Kaluza-Klein monopoles in M-theory, while
they also arise in the quantum moduli spaces of N=4 supersymmetric gauge field
theories with SU(2) gauge group and matter hypermultiplets in three spacetime
dimensions. We comment on rotational isometries versus translational isometries
in the context of N=2 NLSM in terms of projective hypermultiplets.
