  We show that $3$-Sasaki structures admit a natural description in terms of
projective differential geometry. This description provides a concrete link
between $3$-Sasaki structures and several other geometries and constructions
via a single unifying picture. First we establish that a $3$-Sasaki structure
may be understood as a projective structure equipped with a certain holonomy
reduction to the (possibly indefinite) unitary quaternionic group
$\textrm{Sp}(p,q)$, namely a parallel hyperk\"ahler structure on the projective
tractor bundle satisfying a particular genericity condition. For the converse,
where one begins with a general parallel hyperk\"ahler structure on the
projective tractor bundle, the genericity condition is not automatic. Indeed we
prove that generically such a reduction decomposes the underlying manifold into
a disjoint union of strata including open manifolds with (indefinite)
$3$-Sasaki structures and a closed separating hypersurface at infinity with
respect to the $3$-Sasaki metrics. Moreover, it is shown that the latter
hypersurface inherits a Biquard-Fefferman conformal structure, which thus
(locally) fibres over a quaternionic contact structure, and which in turn
compactifies the natural quaternionic K\"ahler quotients of the $3$-Sasaki
structures on the open manifolds. As an application we describe the projective
compactification of (suitably) complete, non-compact (indefinite) $3$-Sasaki
manifolds and recover Biquard's notion of asymptotically hyperbolic
quaternionic K\"ahler metrics.
