  We define and study an extended hyperbolic space which contains the
hyperbolic space and de Sitter space as subspaces and which is obtained as an
analytic continuation of the hyperbolic space. The construction of the extended
space gives rise to a complex valued geometry consistent with both the
hyperbolic and de Sitter space. Such a construction shed a light and inspires a
new insight for the study of the hyperbolic geometry and Lorentzian geometry.
We discuss the advantages of this new geometric model as well as some of its
applications.
