I have noticed recently some purple dust-like material on the top of
my compost. I pour enough liquid on, that it should wash off, but the
fuzz is there again a couple of days later. So I am guessing that
this a some kind of fungus, mold, or something like that.

Sounds like it.

I don't know much about identifying one kind versus another, but a
compost bin is all about letting the decomposers (fungus, bacteria,
etc) do their thing. So the chances are quite good that this is a
non-problem (and in fact will stop once the decomposition gets to a
certain point).