In number theory, an adele is a restricted product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular points, this completion is a ring of formal power series. The adeles are intuitively defined so that we have an injective homomorphism from the ring of functions on a given open set (Zariski open for a number field) to the adeles for that open set.
My question is, what if we replace formal power series by convergent Laurent series (i.e. in the complex topology)? This is what I might call a holomorphic adele. There is a natural injective map from the ring of meromorphic functions on an open set to the ring of adeles on that set. We can also put a topology on these "holomorphic adeles" similar to how it's normally done. Is this construction ever considered? Might it be useful? Is there a connection between the adelic topology and the complex topology?