If you are interested in dialogue between Christianity and Buddhism, look up Thomas Merton and DT Suzuki. With monasticism and deep prayer, much of the same ideas and imagery may be employed. There are parallels, but also sincere differences.
Notably, Buddhist ideas of Self is that it is Sunyata, which is often translated as Void. The idea is that the Self doesn't really exist, as at heart there is only the One True Buddha, a sort of Monism. So by attaining enlightenment, the Self dissipates into the unity of Non-Duality, where there is both Self and Not-Self. The Self is largely a confusion of the Khandas, or heaps of desire, that flows forth - constantly changing like a river, and like a river a sort of constant, but the water is different the whole time.
In Mahayana Buddhism, the ultimate end is to help all beings attain enlightenment, and as the Self can only differentiate itself from the One True Buddha Nature by use of a subdividing mind, when that happens, all will simply be the Monistic One in an ultimate state of non-duality - so the self and all bodhisattva will cease to exist. Therevada is a bit different.
Also, Buddhism talks a lot about compassion, but Buddhist compassion is not the Christian variety. Buddhism teaches that desire and attachment leads to suffering, so the goal is to lead all to have no desires tying them here. So Buddhist compassion for suffering also entails compassion for 'suffering' that we would not see as such - such as devotion to a spouse, or love of your children - as ultimately that will pass away and result in pain. This is why Buddhism can excuse Gautama abandoning his wife and child. A noted difference in concept exists between what we mean.
Another good point, is that the One True Buddha is often equated to God by many - but it is more an monistic unity, than a personal relationship. So Buddhists will say things like the love of Oya-sama or the Pure Land of Amidha, that sound Christianesque, but read deeper and the underlying worldview is quite different. A good way to see this is to read up a bit about Nagarjuna at Nalanda, which is where much Mahajana comes from. Keep in mind that there are quite marked differences between the three schools of Buddhism, the three turnings of the wheel of the Dharma of Mahayana, Theravada and Vajrayana, so I am generalising quite a bit.
I read a bit about Buddhism a few years ago, back when trying not to be a Christian, and while I have a lot of sympathy for its monism and the almost Stoic Apatheia in much Mahayana, I could never shake the feeling that Attachment is essentially Good. That loving something is Good, even if temporary and may result in pain. In my mind, the failure of Buddhism in the West in Greco-Roman times goes back to the bedrock of Plato, that essentially optimistic idea of Love from the Symposium that also played such a part in Christianity. Many of the similarities are false friends, only superficially compatible, and I think the modern cachet of Buddhism in the West is mostly the same type of superficial orientalism of Beatles visiting an Indian Guru. When the West falls, maybe things will change, but we are too different culturally for much synthesis.
