Paramatman
Paramatman/Parabhraman or Paramātmā is the Absolute Atman, or supreme Self, in various philosophies such as the Vedanta and Yoga schools in Hindu theology, as well as other Indian religions like Sikhism. The Paramatman is the "Primordial Self" or the "Self Beyond" who is spiritually practically identical with the Absolute, identical with the Brahman. Selflessness is the attribute of Paramatman, where all personality/individuality vanishes.
Etymology
The word stem paramātman is formed from two words, parama, meaning "supreme" or "highest", and ātma, which means individual self.The word "Ātman" generally denotes the Individual Self, but by the word "Paramatman" which word also expresses Boundless Life, Boundless Consciousness, Boundless Substance in Boundless Space, is meant the Atman of all atmans or the Supreme Self or the Universal Self. The word "Atman" which literally means non-darkness or light, is Brahman the subtlest indestructible Divine existence. The word "Paramatman" refers to the Creator all.
Jainism
In Jainism, each atman or individual self is a potential Paramatman or God, both are essentially the same. It remains as atman only because of its binding "karmic" limitations, until such time as those limitations are removed. As Paramatman, the atman represents the ultimate point of spiritual evolution.Even though Jain mysticism centers around Atman and Paramatman because it believes in the existence of soul, in Jainism, which accepts neither Vedic authority nor Monism, all enlightened souls are referred to as Paramatman and regarded as gods. Jainism honours the soul of each man as its own eternally distinct savior. Since the Paramatman of Jainism is unable to create and govern the world, there is no place of God as a creator and bestower of fortune.
Buddhism
Buddhism rejects a metaphysics of "ground" such as the paramatman.Hinduism
Hindus conceptualize the Para Brahman in diverse ways. In the Advaita Vedanta tradition, Nirguna Brahman is Para Brahman. In Dvaita and Vishistadvaita Vedanta traditions, Saguna Brahman is Para Brahman. In Vaishnavism, Shaivism and Shaktism, Vishnu, Shiva and Shakti respectively are Para Brahman. Mahaganapati is considered as Para Brahman by the Ganapatya sect.Description in the Upanishads
The sage of the Brihadaranyaka Upanishad IV.4.2, though not using the word "Paramatman", explains that at the time of release the portion of the Paramatman and the portion of the Jiva presiding in the right eye become unified with the Paramatman and the Jiva presiding in the heart, then the Jiva does not see, smell, taste, speak, hear, feel, touch and know; when Paramatman goes out, the Chief Prana goes out after Him, followed by the Lower Prana. Paramatman goes out riding on the Jiva following consciousness and work, knowledge of former life or natural capacity. In the Prashna Upanishad IV.11 the word "Atman" cannot refer to Jiva because the Jiva cannot of its own accord throw off its body or understand avidya, therefore, it refers to Paramatman. The Jiva attains Moksha when he actually knows the Paramatman, the Asarira Prajnatman, to be thousand-headed, to be the governor of all and to be superior to all. Thus, Paramatman is one of the many aspects of Brahman and has all attributes of Brahman. Atman and Paramatman are one, some say they are distinct as well as one, they are one with reference to Shakti but distinct with reference to that power.Parable of the Two Birds
The word Paramatman is not to be found in the Rig Veda but through allusion referring to Paramatman as Isha. This distinction is made because all of its mantras which in the form of prayers are addressed to gods. In its great Riddle Hymn is the famous mantra - R.V.I.164.20, that was revealed to Rishi Deergatamaah Auchathyah and borrowed by Mundaka Upanishad III.1.1-3, which belongs to Atharva Veda, to weave the parable of the Two Birds:-Two birds.Aurobindo makes the Spirit or Purusha the Source of everything, including Brahman. He makes Purusha more fundamental. Thus, he does not have to say Brahman to be the source of inferior Brahman, and he also dismisses the sense of Reality revealed in imaginative and emotional build-up.
Case of Two souls
The Dualistic School of Philosophy initiated by Anandatirtha draws its support from the afore-cited passage as well as from the passage of Katha Upanishad I.3.1 of an earlier Upanishad that speaks about two souls which taste the fruits of action, both of which are lodged in the recess of the human heart, and which are different from each other as light and shade, that carried the flaw—how could the Universal soul be regarded as enjoying the fruits of action? The followers of Madhava draw their support from the Bhagavad Gita XV.16 that speaks about two persons in this world, the Mutable and the Immutable; the Mutable is all these things, while the Immutable is the one who exists at the top of them, one is the Jivatman and the other, Paramatman. Jivatman is chit, the sentient, and Paramatman is Isvara, both have the same attributes; they are inseparably present together on the tree which is achit, the insentient, or the gross Avidya component of existence. Jivatman and Paramatman are both seated in the heart, the former is driven by the three modes of nature and acts, the latter simply witnesses as though approving the former's activities. The relationship between Paramātmā, the Universal Self, and 'ātma, the Individual Self, is likened to the indwelling God and the soul within one's heart. Paramatman is one of the many aspects of Brahman. Paramatman is situated at the core of every individual jiva in the macrocosm. The Upanishads do compare Atman and Paramatman to two birds sitting like friends on the branch of a tree where the Atman eats its fruits, and the Paramatman only observes the Atman as a witness of His friend's actions.Advaita
In Advaita philosophy, individual souls are called Jīvātman, and the Highest Brahman is called Paramātman. The Jivatman and the Paramatman are known to be one and the same when the Jivatman attains the true knowledge of the Brahman. In the context of Advaita, the word Paramatman is invariably used to refer to Nirguna Brahman, with Ishvara and Bhagavan being terms used to refer to Brahman with qualities, or Saguna Brahman.Brahman and Isvara are not synonymous words, the apparent similarity is on account of similar looking attributes imagined with regard to the impressions these two words activate. According to Advaita, Isvara is Brahman associated with Maya in its excellent aspect, as the empirical reality it is the determinate Brahman; Isvara has no reality apart from Brahman. The Svetasvatara Upanishad developed the conception of a personal God. The Katha Upanishad states that never has any man been able to visualise Paramatman by means of sight, heart, imagination or mind. The Anandamaya-kosha is the Isvara of the Upanishads. Gaudapada called duality Maya, and non-duality, the only reality. Maya is the Cosmic Nescience that has in it the plurality of subject and object and therefore, Isvara is organically bound with the world. Beyond the Prana or Isvara is the state of the Infinite limitless Brahman which is why in the Bhagavad Gita VII.24, Krishna tells Arjuna—"not knowing My unsurpassable and undecaying supreme nature the ignorant believe Me to have assumed a finite form through birth."
With regard to the cause of Samsāra, as to where it resides and the means of its removal, Adi Shankara in his Vivekachudamani.49. instructs that the individual self is the Paramatman in reality, the association of the individual self with ajnana i.e. with avidya, which he terms as anatmabandhah, bondage by the anatman or non-atman, makes it to identify itself with gross, subtle and causal bodies and from that arises Samsāra which is of the form of superimposition of qualities of sukha, dukha etc., on itself, the atman.
Vaishnavism
Paramatman is beyond knowledge and ignorance, devoid of all material attributes. In Chapter 13 of the Bhagavad Gita, Paramatman is described as Krishna residing in the hearts of all beings and in every atom of matter. He is the overseer and the permitter of their actions. Paramatman is different from five elements, the senses, mind, pradhana and jiva.Vaishnava sects maintain that attaining knowledge of Brahman and identification of Atman with Brahman is an intermediate stage of self-realization, and only Bhakti Yoga can lead to the next step of Paramatman realization as the indwelling God, ultimately leading up to liberation by God-realization.
The Viṣṇu or the deity of the quality of goodness in the material world is the puruṣa-avatāra known as Kṣīrodakaśāyī Viṣṇu or Paramātmā.
In Bengal, Vaishnava Krishna is viewed as one endowed with his essential svarupa-shakti; he is Bhagawat in full manifestation endowed with Jivasakti and Mayasakti, he the Paramatman and Brahman. Brahman, Paramatman and Bhagavan are 3 gradations of the ultimate reality.