Quaternions as a counterexample to the Gelfand–Mazur theorem

Question

It seems that by the Gelfand–Mazur theorem quaternions are isomorphic to complex numbers. That is clearly wrong. So where is the catch?

I think that I found the problem but it seams so subtle that would like to have confirmation from someone else and I would like to know where the proof of Gelfand–Mazur blows up.

Answer 1

The quaternions are certainly a real Banach algebra, but are you sure they are a complex Banach algebra? If you identify $i$ with $i$ to get the obvious action, the scalar multiplication doesn't seem to be bilinear . . .