  Fuzzy hyperboloids naturally emerge in the geometries of D-branes, twistor
theory, and higher spin theories. In this work, we perform a systematic study
of higher dimensional fuzzy hyperboloids (ultra-hyperboloids) based on
non-compact Hopf maps. Two types of non-compact Hopf maps; split-type and
hybrid-type, are introduced from the cousins of division algebras. We construct
arbitrary even-dimensional fuzzy ultra-hyperboloids by applying the Schwinger
operator formalism and indefinite Clifford algebras. It is shown that fuzzy
hyperboloids, $H_F^{2p,2q}$, are represented by the coset, $H_F^{2p,2q}\simeq
SO(2p,2q+1)/U(p,q)$, and exhibit two types of generalized dimensional
hierarchy; hyperbolic-type (for $q\neq 0$) and hybrid-type (for $q=0$). Fuzzy
hyperboloids can be expressed as fibre-bundle of fuzzy fibre over hyperbolic
basemanifold. Such bundle structure of fuzzy hyperboloid gives rise to
non-compact monopole gauge field. Physical realization of fuzzy hyperboloids is
argued in the context of lowest Landau level physics.
