If return a = return b then does a=b?

Question

Can you prove that if return a = return b then a=b? When I use =, I mean in the laws and proofs sense, not the Eq class sense.

Every monad that I know seems to satisfy this, and I can't think of a valid monad that wouldn't (Const a is a functor and applicative, but not a monad.)

Answer 1

No. Consider the trivial monad:

data Trivial a = Cow

instance Monad Trivial where
  _ >>= _ = Cow
  return _ = Cow