  We consider supersymmetric quantum mechanics on a K\"{a}hler cone, regulated
via a suitable resolution of the conical singularity. The unresolved space has
a $\mathfrak{u}(1,1|2)$ superconformal symmetry and we propose the existence of
an associated quantum mechanical theory with a discrete spectrum consisting of
unitary, lowest weight representations of this algebra. We define a
corresponding superconformal index and compute it for a wide range of examples.
