The policies will vary from hospital to hospital, and depends on each individual situation. A C-section birth generally requires a longer stay than a vaginal delivery. If there are any complications with the mother or her baby, then the possibility of an elongated stay is also likely.
In most hospitals, new mothers have the option of keeping the baby with them or having them stay in the nursery in order to get more rest. A combination of the two would probably work best since just having gone through labour will be strenuous for mum, but this is up to the preferences of the parents.
Regardless of when you get discharged, you will need another adult to accompany you and a car seat appropriate for your baby before you will be released. Many think of the delivery hospital stay as a requirement to ensure that mum and her baby are safe, but it can also be a welcome respite about the tasks of labour and childbirth. It is also a great opportunity to get used to being a mother while surrounded by experts.