Starting from the Zariski topology, a
natural notion of nonstandard generic point is introduced in
complex algebraic geometry. The existence of this kind of point is
a strong form of the Nullstellensatz. This notion is connected
with the classical concept of generic point in the spectrum
$\text{Spec}({\cal A}_{n,\mathbb C})$ of the corresponding algebra
${\cal A}_{n,\mathbb C}$. The nonstandard affine space
${^*\mathbb C}^n$ appears as an affine unfolding of the geometric
space $\text{Spec}({\cal A}_{n,\mathbb C})$. This affine space is
the disjoint union of the sets whose elements are the nonstandard
generic points of prime and proper ideals of ${\cal
A}_{n,\mathbb C}$: this structure leads to the definition of
algebraic points in ${^*\mathbb C}^n$. A natural extension to
analytic points in ${^*\mathbb C}^n$ is given by Robinson's
concept of generic point in local complex analytic geometry. The
end of this paper is devoted to a generalization of this point of
view to the real analytic case.