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. They show this for representations of projective varieties that are cohomologically rigid, which holds for complex hyperbolic manifolds by Mostow rigidity. However, there are hyperbolic 3-manifolds which have non-integral traces, implying that they cannot embed isometrically in a complex hyperbolic manifold.