How applicable is the phrase “I am God” to Hinduism?
If you want to know whether the phrase "I am God" has validity in Hinduism, that exact phrase is present in the Brihadaranyaka Upanishad (Nikhilananda's translation):
They say: “Since men think that by the Knowledge of Brahman they become all, what, pray, was it that Brahman knew by which It became all?”
This [self] was indeed Brahman in the beginning. It knew itself only as “I am Brahman.” Therefore it became all. And whoever among the gods had this enlightenment, also became That [Brahman]. It is the same with the seers (rishis), the same with men. The seer Vāmadeva, having realized this [self] as That, came to know: “I was Manu and the sun.” And to this day, whoever in a like manner knows the self as “I am Brahman,” becomes all this [universe]. Even the gods cannot prevent his becoming this, for he has become their Self.
(Note that Brahman is only one possible translation of the word "god"; see my answer in Meta.) Now the question becomes how to interpret the passage. Some might interpret it as saying that the only thing that's keeping you from "becoming the universe" is that you don't realize that you're already the Universe. Others would say, however, that the passage means that you used to be Brahman and that you can become Brahman after you realize your former state, but that you're not Brahman right now.
The Hindu religion doesn't have a single position about the relationship between Jivatma, the individual soul, and Paramatma, the supreme soul or soul of the gods. Rather, different sects of Hinduism, like Advaitam, Dvaitam, and Visishtadvaitam, have differing views on the question.
According to Adi Shankaracharya's philosophy of Advaitam, Jivatma is already identical to Paramatma, he just doesn't realize it yet because he's in the illusion of Maya. Madhvacharya's philosophy of Dvaitam, on the other hand, posits that Jivatma and Paramatma are totally seperate, and that the best the Jivatma can hope for is to get into more elevated state, never actually becoming one with Paramatma, as I discuss in this answer. And Ramanujacharya's philosophy of Vishistadvaitam (the philosophy of Sri Vaishnavas like myself) advocates a compromise position, where there is larger whole but within that whole there is some division, akin to the relationship between a fetus and a pregnant woman; the fetus is a part of the pregnant woman's body, yet it's not the same as the pregnant woman herself.
And there are other philosophies like Chaitanya Mahaprabhu's Achintya Bheda Abheda, where there is simultaneous oneness and difference, but where the coexistence of unity and plurality is beyond Man's comprehension.
So to sum up, while the phrase "I am Brahman" does occur in Hindu scripture, the larger question of whether you are Brahman right now is one that is hotly debated amongst the different sects.
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