All valuations of $\mathbb{Q}$ are associated to either a p-adic field or the field of real numbers. Now, the classification of fields gives that every local field in characteristic $0$ is a finite extension of these.
Is there another notion for a topology on $\mathbb{Q}$ than coming from a valuation, which allows more general completions? Preferably, their additive groups should have Haar measures, but are not necessarily locally compact?