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 conjecturs by Konsevich and Soibelman, as well as positive results by various authors (Berkovich, Nicaise, Boucksom, Jonsson...).
In a joint work (to be released 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 focused on this construction and its basic properties, 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.