The hill range that separates the state of Manipur from the state of Nagaland is known as the Barail hills.

Option 1, Arakan hills, is incorrect as the Arakan hills are located in the southeastern part of Bangladesh and south of the Chin State in Myanmar.

Option 2, Patkai hills, is incorrect as the Patkai hills are a mountain range in northeastern India and northern Myanmar, and they do not separate Manipur and Nagaland.

Option 4, Manipur, is incorrect as Manipur is a state in northeastern India and not a hill range.

Therefore, option 3, Barail hills, is the correct answer. The Barail hills are a range of low hills that stretch between Manipur and Nagaland, forming a natural boundary between the two states. They are an important feature of the northeastern geography and play a role in determining the cultural and ecological diversity of the region.