The authors construct a family of quaternionic Kähler structures on the tangent bundle of any complex space form of complex dimension at least 3. The quaternionic Kähler structure depends on the choice of a positive function on the half-line, satisfying some bounds. All these structures are hyper-Hermitian and of cohomogeneity one, but are in general not hyper-Kählerian. An exception is Calabi's famous hyper-Kähler metric on the (co-) tangent bundle of complex projective space. The hyper-Hermitian structures discussed here belong to a wider class of almost hyper-Hermitian structures on the tangent bundle of an almost Hermitian manifold, which were constructed by the authors [\textit{M. Tahara, S. Marchiafava} and \textit{Y. Watanabe}, Note Mat. 18, 131-141 (1998; Zbl 0964.53021)].