  On asymptotically complex hyperbolic (ACH) Einstein manifolds, we consider a
certain variational problem for almost complex structures compatible with the
metric, for which the linearized Euler-Lagrange equation at K\"ahler-Einstein
structures is given by the Dolbeault Laplacian acting on $(0,1)$-forms with
values in the holomorphic tangent bundle. A deformation result of Einstein ACH
metrics associated with critical almost complex structures for this variational
problem is given. It is also shown that the asymptotic expansion of a critical
almost complex structure is determined by the induced (possibly non-integrable)
CR structure on the boundary at infinity up to a certain order.
