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This is not exactly an answer (but maybe too long for a comment). Essentially, the only semi-Riemannian submersion from a complex pseudo-hyperbolic space onto any 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.