  Conformal transformations of a Euclidean (complex) plane have some kind of
completeness (sufficiency) for the solution of many mathematical and
physical-mathematical problems formulated on this plane. There is no such
completeness in the case of Euclidean, pseudo-Euclidean and polynumber spaces
of dimension greater than two. In the present paper we show that using the
concepts of analogical geometries allows us to generalize conformal
transformations not only to the case of Euclidean or pseudo-Euclidean spaces,
but also to the case of Finsler spaces, analogous to the spaces of affine
connectedness. Examples of such transformations in the case of complex and
hypercomplex numbers H_4 are presented. In the general case such
transformations form a group of transitions, the elements of which can be
viewed as transitions between projective Euclidean geometries of a
distinguished class fixed by the choice of metric geometry admitting affine
coordinates. The correlation between functions realizing generalized conformal
transformations and generalized analytical functions can appear to be
productive for the solution of fundamental problems in theoretical and
mathematical physics.
