There exist hyperbolic 3-manifolds which cannot embed totally geodesically in complex hyperbolic manifolds, answering this question in the negative.
Recently it was shown by Esnault-Groechenig that complex hyperbolic manifolds have integral discrete faithful representations, although this is not stated directly in their paper.
They show that cohomologically rigid representations of the fundamental group of smooth projective varieties must be integral. Compact hyperbolic $n$-manifolds are projective varieties, and the discrete faithful representation into $SU(n,1)$ is unique up to conjugation and cohomologically rigid by Mostow rigidity. Hence this representation must have integral traces (this corollary was pointed out to me by David Fisher).
However, there are hyperbolic 3-manifolds such that the discrete faithful representation of the fundamental group into $SO(3,1)$ has non-integral traces, implying that they cannot embed isometrically in a complex hyperbolic 3-manifold.