Maybe not allergies
A 2 month old baby with "dark purplish-blueish bags" under the eyes is very young, so maybe doesn't have allergies
as the cause.
Because the baby will have seen the doctor for regular check-ups at ages 1m, 2m, 4m, and 6m, plus will see the doctor again at 9m, there are many opportunities for the parents to get the doctor's opinion. Also, the trait of darker lower lids' skin runs in some families.