Is Balarama an Avatar of Vishnu or Adiseshan (or both)?


The reason that Balarama is called an incarnation (avatara) of Vishnu is that the serpent Adiseshan is himself a form of Vishnu. Here is how Balarama's birth is described in the Srimad Bhagavatam:

When Kamsa had killed six of the children born from Devakî, the seventh one, a plenary expansion of Vishnu who was celebrated with the name Ananta, therefore [was born] as an embryo in the womb of Devakî ... [Vishnu] instructed His spiritual potency [Yoga-mâyâ] as follows: ... "In the womb of Devakî there is the embryo known as [Ananta-]S'esha who is a plenary expansion of Me. Take care of a smooth transition from her womb to the womb of Rohinî.... Because He changes wombs [from Devakī to the womb of Rohinī] the people of the world will address Him with the name Sankarshana, because He brings pleasure to the people [of Gokula] He will be called Râma and because of His great physical strength He will be named Balabhadra."

And here is another passage in the Srimad Bhagavatam that describes the role that Adiseshan plays as a form of Vishnu:

At a distance of thirty-eight thousand yojanas beneath the base of Pâtâla He is situated who, as a part of the Supreme Lord, relates to the darkness and is called Ananta. Scholarly Vaishnavas describe Him as Sankarshana, the ruler of the ego or I that is characterized by self-awareness [pride, identification], because He unites - 'draws together' - the seer and the seen, the subject and the object. The celestial sky around the earth, this universe, sustained on only one of the thousands of hoods of the Supreme Lord in the form of Anantadeva, is seen as a [tiny] white mustard seed.

Finally, in the Bhagavad Gita Krishna says "Among the Nagas I am Ananta."

On a side note, since you were wondering, Balarama reverting back into Adiseshan is described in this chapter of the Mausala Parva of the Mahabharata:

Proceeding then to the forest, Keshava [i.e. Krishna] beheld Rama sitting in a solitary spot thereof. He also saw that Rama had set himself to Yoga and that from out his mouth was issuing a mighty snake. The colour of that snake was white. Leaving the human body (in which he had dwelt so long), that high-souled naga of a 1,000 heads and having a form as large as that of a mountain, endued besides with red eyes, proceeded along that way which led to the ocean.

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