There are many proofs that there is a unique division quaternion algebra over a locally compact field that is not $\mathbb{C}$. For instance this set of notes/book by John Voight contains two proofs: http://www.cems.uvm.edu/~voight/crmquat/book/quat-modforms-041310.pdf
That said, every proof that I've seen first splits the proof into the archimedean and non-archimedean cases (and then sometimes into the residue characteristic 2 and $\ne$ 2 cases).
Even Weil's Basic Number Theory doesn't seem to have a proof that treats all places equally (maybe I missed it?). Does anyone know of such a proof?