There is a general slogan according to which the limit behaviour of a one-parameter family of complex algebraic varieties when the parameter /t/ tends to zero should be (partially) encoded in the associated /t/-adic analytic space in the sense of Berkovich; this slogan has given rise to deep and fascinating conjectures by Konsevich and Soibelman, as well as positive results by various authors (Berkovich, Nicaise, Boucksom, Jonsson...).

In a joint work (hopefully to be relaeased soon!) with E. Hrushovski and F. Loeser, we develop a new approach to that kind of question. It consists in building, using ultraproducts, a field which caries both a "non-standard complex structure" and a non-archimedean /t/-adic structure. My talk will be mainly focussed on this construction and its basic properties, and especially on integration over this field of non-standard complex numbers. But I will also perhaps say a few words about the way we apply it to describe in purely non-Archimedean terms the limit of certain complex integrals when /t/ tends to zero.