Avatars of Vishnu were a well established concept
Jesus Christ is God, the incarnation of the Second Person of the Trinity, which is Love. He is not an avatar in the Hindu sense, although such points are difficult to argue since there is no equivalent to the Magisterium in Hinduism.
For the most part, although some may say that actual persons, such as Mohammed, were avatars, they are essentially mythical entities in stories that are meant to demonstrate the restoration of a moral order, the Dharma, when the world has gone too astray, They are not historical beings, but rather speak to the ontological structure of reality - the world of the Spirit, as it is understood by eastern cultures and traditions.
Jesus is the Living Way to God. We surrender ourselves to Him, becoming Love itself, on our journey, destined for communion, in and through Him, with the Triune Godhead. His body and blood are the Eucharist, which we ingest to further this process, that ultimately rests on our actions, and His grace, Together, in Love, in giving of ourselves to God and one another, we form the mystical body of Christ, the true vine, wherein we are One, while maintaining our indiviual being.
It could be said that what Hinduism describes in its mythology, finds its reality in Jesus and all who have taken Him into their hearts, expressing the Love that is God. We can all be avatars then, and Atman would be truly Brahman, where reality is grounded in Perfect Relationality as Divine Love.