Can I keep my original citizenship when I become Norwegian?

Norway permits dual citizenship, so becoming Norwegian does not in itself require you to give up your existing nationality. When you apply, you do not need to tell UDI separately that you wish to keep your other citizenship.

Whether you actually keep the other nationality also depends on the laws of your home country, since some countries automatically withdraw citizenship when you naturalise elsewhere. It is sensible to confirm your home country's rules independently.