  We use twistor methods to promote Namikawa's universal Poisson deformations
of conic affine symplectic singularities to families of hyperk\"ahler
structures deforming hyperk\"ahler cone metrics. The metrics we produce are
generally incomplete, but for specific classes of hyperk\"ahler cones, even
these incomplete metrics have some interest and applications: we study in
detail the case of the nilpotent cone of a simple complex Lie algebra, with
applications to hyperk\"ahler metrics with symmetries and hyperk\"ahler
quotients, and the case of Kleinian singularities, with applications to
codimension-4 singularities of G2-holonomy metrics and their dual description
in theoretical physics in terms of 3-dimensional gauge theory.

Differential Geometry;High Energy Physics - Theory;Algebraic Geometry