This was suggested by a question in quora.
It is well known that any construction that can be done with ruler and compass can be done by compass alone.
Can any additional points be constructed if more than one compass can be used?
I have no idea how to answer this and I do not even know how multiple compasses would be used in a construction.
Your intuition ("How would one even use multiple compasses?") is correct. Think how you use the compass. You make arcs of arbitrary or fixed radii, starting from arbitrary or fixed points (usually the former at the beginning of the construction, usually the latter later).
If you were to not use the compasses simultaneously, then any time you used Compass B, you could substitute Compass A for it instead. (Assume for the sake of argument our compasses don't wear out from overuse.) So any construction that required both compasses would require them to be used simultaneously.
So consider under what circumstances it would be necessary to draw two arcs simultaneously. You would only need to do this if one of the arcs (say that drawn by Compass B) for some reason could not be drawn later. The only reason this could happen is if the radius or the center of this arc could not be recorded to be drawn later by Compass A. If the center and radius of this arc were determined by preceding steps in this construction, then they could be retrieved and the arc drawn by Compass A. So either the radius or the center of this arc must be arbitrary. But if the radius is arbitrary and the center fixed, you could always make an arc of random radius at that fixed center later, and similarly if only the center were arbitrary or both were arbitrary.
In conclusion, there is nothing you can do with two compasses you cannot do with one.