  We provide a new general scheme for the geometric quantisation of
$\operatorname{Sp}(1)$-symmetric hyper-K\"ahler manifolds, considering Hilbert
spaces of holomorphic sections with respect to the complex structures in the
hyper-K\"ahler 2-sphere. Under properness of an associated moment map, or other
finiteness assumptions, we construct unitary quantum (super) representations of
central extensions of certain subgroups of Riemannian isometries preserving the
2-sphere, and we study their decomposition in irreducible components. We apply
this quantisation scheme to hyper-K\"ahler vector spaces, the Taub--NUT metric
on $\mathbb{R}^4$, moduli spaces of framed $\operatorname{SU}(r)$-instantons on
$\mathbb{R}^4$, and partly to the Atiyah--Hitchin manifold of magnetic
monopoles in $\mathbb{R}^3$
