  We demonstrated that classical mechanics have, besides the well known quantum
deformation, another deformation -- so called hyperbolic quantum mechanics. The
classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the
ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus
there are two different deformations of classical phase-space: complex Hilbert
space and hyperbolic Hilbert space (module over a so called hyperbolic algebra
-- the two dimensional Clifford algebra). To prove the correspondence principle
we use the calculus over the hyperbolic algebra similar to functional
superanalysis of Vladimirov-Volovich. Ordinary (complex) and hyperbolic quantum
mechanics are characterized by two types of interference perturbation of the
classical formula of total probability: ordinary $\cos$-interference and
hyperbolic $\cosh$-interference.
