In this article, the author considers the problem of introducing an almost complex structure to a given arbitrary asymptotically complex hyperbolic (ACH) Einstein space that extends the CR structure on the boundary in an appropriate sense and in a canonical manner. The almost complex structures \(J\) that are compatible with a given ACH metric \(g\) are considered, and are extensions of the conformal infinity of \(g\). The author introduces a functional which is defined on the space of those \(J\) for which \((g,J)\) is an ACH almost Hermitian structure. It is obvious that \(J\) is a critical point of the functional if \((g, J)\) is Kähler, or more generally, if \((g,J)\) is semi-Kähler, as called by P. Gauduchon [Math. Ann. 267, 495–518 (1984; Zbl 0523.53059)]. The choice of the functional makes the linearization \(P_S\) of the mapping \(J \to S\) a Laplace-type differential operator. So, if what is given in the beginning is not only a metric but an ACH and an almost Hermitian structure that is Kähler-Einstein (or an ACH Kähler-Einstein structure for short), then under some assumption, one can construct a family of deformed ACH almost Hermitian structures. This assumption is satisfied when \(g\) has negative sectional curvature, for instance. For future applications, knowing the asymptotic expansion of \((g,J)\) is also important. This is also achieved.