This is not exactly an answer (but maybe too long for a comment). 
Essentially, the only semi-Riemannian submersion from a complex hyperbolic space onto a manifold whose fibers are complex, connected, and totally geodesic is the semi-Riemannian submersion from $\mathbb{CH}_1^{2n+1}$ to the quaternionic hyperbolic space $\mathbb{H}H^n$. (This is the twistor space construction over  $\mathbb{H}H^n$).
This is proved by G. Baditoiu and S. Ianus in Theorem 2.8 of [this paper][1].





  [1]: https://arxiv.org/pdf/math/0005228.pdf