Motivated by the quest to better understand the structure of the adèle class space of a global field in characteristic zero, Connes and Consani recently initiated a theory of hyperfield extensions of the so-called ``Krasner hyperfield'' which is one of the realizations of the field with one element. Remarkably, there appears a deep correspondence with certain group actions on certain combinatorial geometries. We will elaborate on brand new results in this theory, make wild speculations and pose several important open questions on the way.