  The $k$-Cauchy-Fueter complex in quaternionic analysis is the counterpart of
the Dolbeault complex in complex analysis. In this paper, we find the explicit
transformation formula of these complexes under ${\rm SL}(n+1,\mathbb{H})$,
which acts on $\mathbb{H}^{ n}$ as quaternionic fractional linear
transformations. These transformation formulae have several interesting
applications to $k$-regular functions, the quaternionic counterpart of
holomorphic functions, and geometry of domains. They allow us to construct the
$k$-Cauchy-Fueter complex over locally projective flat manifolds explicitly and
introduce various notions of pluripotential theory on this kind of manifolds.
We also introduce a quaternionic projectively invariant operator from the
quaternionic Monge-Amp\`{e}re operator, which can be used to find projectively
invariant defining density of a domain, generalizing Fefferman's construction
in complex analysis.
