Artist’s depiction of the Hindu concept of Atman; Aang and Korra's Cosmic Selves
In Hindu philosophy, Atman refers to the innermost essence of an individual. It lies beneath and guides all the activities of a person just as Brahman, the Absolute All, underlies the workings of the whole universe.
When Aang encounters his Atman, it is during the act of unlocking his Crown Chakra and relinquishing all worldly attachments.
Korra, however, achieves enlightenment when she connects with the cosmic energy of the universe, and in doing so uncovers the fundamental truth of Atman - it is both part of and identical to Brahman. Thus, she is able to project her Atman into the physical plane of existence, having become a master of time and space.