You can start a new Apple ID, just understand all purchases made under an Apple ID are forever associated with that ID and can't be switched to a new one.
The requirements of an Apple ID is that it be from a valid email account.
http://support.apple.com/kb/HT2204
Since many apps are free, they can be repurchased free with a new Apple ID.
If you purchased valuable apps under your ID, get a new one for your daughter. Or visa versa.
You can continue to keep them in one computer library which will allow you to sync them to all devices.
Updates will require the Apple ID and password from the purchasing Apple ID.
Changing Apple ID (new email address )
http://support.apple.com/kb/HT5621