The function fields (in one or more variables) over $\mathbb{F}_q$ are precisely the infinite, finitely generated fields of characteristic $p$. Thus an at least reasonable characteristic $0$ analogue is given by the (necessarily infinite!) finitely generated fields of characteristic $0$. In other words, function fields in finitely many (possibly zero) variables over a number field $K$.

Indeed there has been much work on generalizing arithmetic geometric statements over global fields to arithmetic geometric statements over arbitrary infinite, finitely generated fields. The one which springs most readily to my mind is the following generalization of the Mordell-Weil theorem due to Lang and Neron: the group of rational points on any abelian variety over any finitely generated field is a finitely generated abelian group.