Most people after awakening recognise their primary essence, Consciousness. They also recognise that they have an egoic mind or mind made self which creates unwanted thinking.
This thinking could be imaginative or drawing on past memories and emotional pain. They can be judgemental thoughts, fearful, lustful, greedy, prideful, shameful. Repetitive, negative, energy consuming.
I call this a bunch of diferent names in the book; The Mind Made Self, The Ego Mind, The Unhappy Mind. We could even call it our individuated soul mind, the little ‘i’.
True freedom is said to be realised when this little ‘i’ is completely absorbed into the big ‘I’ of the Self. All the mind tendencies, the entire blueprint of the mind is dissolved into Consciousness.
One is then said to be a JivanMukta – Liberated Whilst Living.
One still has the capacity to use ones mind and creative thoughts, yet nothing is held onto. One lives in perfect contentment with peace of mind at all times.
Ramana Maharshi used to advise to focus on one’s ‘cave of the heart’ and watch for the ‘i’ thought arising. Eventually it can become completely absorbed.
Shiva Rudra Balayogi advises to focus ones mind and eyeballs at the point between ones eyebrows. Just observe there and over time the mind tendencies will evaporate and eventually with grace the mind can completely become absorbed.
A decent Wiki article on the subject can be found here: https://en.wikipedia.org/wiki/Jivanmukta