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Hyperbolic hypercomplex D-bar operators, hyperbolic CR-equations and harmonicity

This paper is partially a review of the development of the Investigation Program announced by Stancho Dimiev at the Bedlevo Conference on Hypercomplex Analysis (2006). A new aspect related with hyperbolic complex numbers, their generalizations and applications on hyperbolic harmonicity is included in some Appendix of S. Dimiev. The recent results on the hyperbolic complex structures are exposed. We describe systematically the $\partial$-bar trik which led to Cauchy-Riemann type systems for different generalized hypercomplex algebras. This is in fact an elementary $\partial$-bar type theory of the PDE on the mentioned algebras. Microlocal treatment (distributions and Fourier transform) of the solutions is included too, with one result of P. Popivanov.