'Pursue' and 'chase' have more or less the same meaning, but 'pursue' is more formal. You can pursue a goal, a dream or a career, for example. 'Chase' suggests a certain speed and urgency - for example, a car chase is when one car is trying to catch up with the car in front of it. 'Hunt' means 'look for', especially when the thing is hidden or hard to find. Detectives hunt for criminals, animals (and people) hunt other animals to kill, shoppers might hunt for bargains in the sales. If you are 'hunting' something, you don't know where it is. If you are 'chasing' it you know where it is - it's ahead of you somewhere, and you are trying to reach it.

They are synonyms. In different contexts, one or other of them may sound more natural to use.