I've heard an FA talking to a GA after a flight about a couple of pax who were apparently awful.. I don't know if anything came of it, but I suppose you could get a note on your permanent record
Im always amazed that FAs usually seem very fond of children, after everything they must put up with.
And for all the people pleading sympathy for the mother, it was her choice to take the baby on the flight in the first place. There are very few situations where a parent HAS to take a baby on a long trip, and I know plenty of people who have limited their travel while their kids are tiny. At a minimum, take them on a shorter flight first so you have an idea of how they'll behave and can avoid longer flights in future if they aren't good flyers. The mother knew what she was getting into, and decided the extra effort was worth it to take her baby on that trip