  We define an index for $\mathfrak{osp}(4^{*}|4)$ superconformal quantum
mechanics on a hyperK\"{a}hler cone. The index is defined on an equivariant
symplectic resolution of the cone, which acts as a regulator. We present
evidence that the index does not depend on the choice of resolution parameters
and encodes information about the spectrum of (semi-) short representations of
the superconformal algebra of the unresolved space. In particular, there are
two types of multiplet which can be counted exactly using the index. These
correspond to holomorphic functions on the cone and to the generators of the
Borel-Moore homology on the resolved space respectively. We calculate the
resulting index by localisation for a large class of examples.
