  For complete complex connections on almost complex manifolds we introduce a
natural definition of compactification. This is based on almost c--projective
geometry, which is the almost complex analogue of projective differential
geometry. The boundary at infinity is a (possibly non-integrable) CR structure.
The theory applies to almost Hermitean manifolds which admit a complex metric
connection of minimal torsion, which means that they are quasi--Kaehler in the
sense of Gray--Hervella; in particular it applies to Kaehler and nearly Kaehler
manifolds. Via this canonical connection, we obtain a notion of c-projective
compactification for quasi--Kaehler metrics of any signature.
  We describe an asymptotic form for metrics that is necessary and sufficient
for c--projective compactness. This metric form provides local examples and, in
particular, shows that the usual complete Kaehler metrics associated to
smoothly bounded, strictly pseudoconvex domains in C^n are c--projectively
compact. For a smooth manifold with boundary and a complete quasi-Kaehler
metric $g$ on the interior, we show that if its almost c--projective structure
extends smoothly to the boundary then so does its scalar curvature. We prove
that $g$ is almost c--projectively compact if and only if this scalar curvature
is non-zero on an open dense set of the boundary, in which case it is, along
the boundary, locally constant and hence nowhere zero there. Finally we
describe the asymptotics of the curvature, showing, in particular, that the
canonical connection satisfies an asymptotic Einstein condition. Key to much of
the development is a certain real tractor calculus for almost c--projective
geometry, and this is developed in the article.
