  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.
