We want to understand how the topology of Berkovich spaces varies when we conjugate the non-archimedean base field. After a short introduction with a discussion of the original problem solved by Serre [1964] in the complex setting, we explain some background material about non-archimedean geometry and non-archimedean analytifications. We are able to construct examples of non-homeomorphic conjugate Berkovich spaces by controlling the homotopy type of the Berkovich analytification via its skeleton and its tropicalization. In the appendix we include some useful programs written in SAGE that compute the examples of the last section.
