Motivated by possible applications to meromorphic dynamics, and generalising known properties of difference-closed fields, this paper studies the theory
CCMA
\operatorname {CCMA}
of compact complex manifolds with a generic automorphism. It is shown that while
CCMA
\operatorname {CCMA}
does admit geometric elimination of imaginaries, it cannot eliminate imaginaries outright: a counterexample to
3
3
-uniqueness in
CCM
\operatorname {CCM}
is exhibited. Finite-dimensional types are investigated and it is shown, following the approach of Pillay and Ziegler, that the canonical base property holds in
CCMA
\operatorname {CCMA}
. As a consequence the Zilber dichotomy is deduced: finite-dimensional minimal types are either one-based or almost internal to the fixed field. In addition, a general criterion for stable embeddedness in
T
A
TA
(when it exists) is established, and used to determine the full induced structure of
CCMA
\operatorname {CCMA}
on projective varieties, simple nonalgebraic complex tori, and simply connected nonalgebraic strongly minimal manifolds.