  We apply methods of nonstandard mathematics in order to regard analytic
geometry in a very different way. For example, complex spaces are seen to be
the "standard part" of certain algebraic nonstandard schemes. We construct a
category of such schemes, sitting in between usual algebraic schemes (over the
complex numbers) and that of complex spaces. We clarify the structure of prime
ideals in a Stein algebra, coming from nonstandard points and show in
particular that ANY maximal and minimal prime ideal in a Stein algebra is the
vanishing ideal of a nonstandard point. Other applications of our point of view
are given for differential forms (a la Leibniz), generic points (as nonstandard
ones), meromorphic functions, hyperbolicity. The essential tools taken from
nonstandard mathematics and adapted for our purposes, are summarized in the
appendix.
