First of all, fundamental groups of compact hyperbolizable 3-manifolds (with or without boundary) are LERF, this is one corollary of the work by Agol, Haglund and Wise. The LERF property is unaffected by passing to finite index subgroups. All torsion-free finitely generated discrete subgroups of $PSL(2,C)$ are fundamental groups of compact hyperbolizable 3-manifolds (follows from Thurston's work and is explained in my book "Hyperbolic Manifolds and Discrete Groups"). Therefore, all finitely generated discrete subgroups of $PSL(2,C)$ are LERF.

As for infinitely generated discrete subgroups, they might not even contain proper finite index subgroups: See Ian Agol's example in Non-residually finite matrix groups and his comment in the end. I did not think about this example for a long time, but at the time when the example was posted I thought about it and concluded that Ian was correct.

First of all, fundamental groups of compact hyperbolizable 3-manifolds (with or without boundary) are LERF, this is one corollary of the work by Agol, Haglund and Wise. The LERF property is unaffected by passing to finite index subgroups. All torsion-free finitely generated discrete subgroups of $PSL(2,C)$ are fundamental groups of compact hyperbolizable 3-manifolds (follows from Thurston's work and is explained in my book "Hyperbolic Manifolds and Discrete Groups"). Therefore, all finitely generated discrete subgroups of $PSL(2,C)$ are LERF.

As for infinitely generated discrete subgroups, they might not even contain proper finite index subgroups: See Ian Agol's example in Non-residually finite matrix groups and his comment in the end. I did not think about this example for a long time, but at the time when the example was posted I thought about it and concluded that Ian was correct.

First of all, fundamental groups of compact hyperbolizable 3-manifolds (with or without boundary) are LERF, this is one corollary of the work by Agol, Haglund and Wise. The LERF property is unaffected by passing to finite index subgroup. All torsion-free finitely generated discrete subgroups of $PSL(2,C)$ are fundamental groups of compact hyperbolizable 3-manifolds (follows from Thurston's work). Therefore, all finitely generated discrete subgroups of $PSL(2,C)$ are LERF.

As for infinitely generated discrete subgroups, they might not even contain proper finite index subgroups: See Ian Agol's example in Non-residually finite matrix groups and the comment in the end. I did not think about this for a long time, but I think that comment was correct.

First of all, fundamental groups of compact hyperbolizable 3-manifolds (with or without boundary) are LERF, this is one corollary of the work by Agol, Haglund and Wise. The LERF property is unaffected by passing to finite index subgroups. All torsion-free finitely generated discrete subgroups of $PSL(2,C)$ are fundamental groups of compact hyperbolizable 3-manifolds (follows from Thurston's work and is explained in my book "Hyperbolic Manifolds and Discrete Groups"). Therefore, all finitely generated discrete subgroups of $PSL(2,C)$ are LERF.

As for infinitely generated discrete subgroups, they might not even contain proper finite index subgroups: See Ian Agol's example in Non-residually finite matrix groups and his comment in the end. I did not think about this example for a long time, but at the time when the example was posted I thought about it and concluded that Ian was correct.