What's wrong with newtypes is that if you want to define "the monoid of integers under addition" and "the monoid of integers under multiplication" (or similar, the second isn't really a monoid because of zero), then an expression like a*x+b becomes some kind of newtype hell:

toAdditive ((toMultiplicative a) <> (toMultiplicative x)) <> toAdditive b

or something similar. This is of course an insane way to program, and so really what is done is that we define two entirely different operations of addition and multiplication so that they can be used without newtypes.

What would be cool is if operations could ad-hoc be bound to instances of typeclasses, for instance you could accumulate a list of integers inside the (0, +) monoid, or inside the (1, *) monoid. Of course this is basically what the fold functions in Haskell do, but you could imagine being able to formalise this pattern at the type level in a less newtype-hellish way.