  In this note we discuss the complex version of the Higgs oscillator on the
hyperbolic space. The eigenvalues and resonances of the complex Higgs
oscillator are computed in different examples in the hyperbolic setting. We
also propose open problems like whether the complex absorbing potential (CAP)
method works for asymptotically hyperbolic manifolds and finding hyperbolic
analogues of the complex harmonic oscillator.
