Atma-Vichara (Self enquiry or Self investigation) is certainly not introspection as in an examination of ones psychological traits, hidden motives, desires, complexes, etc. Nor is it repeating the sentence to oneself, "Who am I?" and trying to formulate an answer. Those may be preliminary practices, but they aren't strictly Atma-Vichara. So what is it? I will let Muruganar tell us in (Ramana approved) translated verse:
To reverse the outgoing nature of the mind and to fix it in the Self, the Heart, one's source, so that it abides there all the time without allowing the ruinous "I"-thought to rise - this is the method of atma-vichara. - Guru Vachaka Kovai, verse 389
Each time a thought arises, do not entertain the idea of allowing it to develop even slightly. [Instead,] the proper course is to first enquire to oneself, "To whom has this thought occured?" - GVK, verse 397
Steve says, "Self inquiry appears to be the trapped in its own questioning dialogue. For me-so far-self -inquiry has not been helpful in meditation. It just leads to more thought and thought is a product of mind; the chatter of the mind. Of course I cannot speak for others but only myself."
The thought "Who am I?", destroying all the other multifarious thoughts, will itself, like the long and sturdy pole used to stir the funeral pyre, finally die, leaving the perfect silence of mauna. GVK, verse 401
Ramana says, "You say 'Who am I?' becomes a japa. It is not meant that you should go on asking 'Who am I?' In that case, thought will not so easily die. In the direct method, as you call it, in asking yourself, 'Who am I?', you are told to concentrate within yourself where the 'I'-thought, the roof of all other thoughts, arises. As the Self is not outside but inside you, you are asked to dive within, instead of going without. What can be more easy than going to yourself? But the fact remains that to some this method will seem difficult and will not appeal. That is why so many different methods have been taught. Each of them will appeal to some as the best and easiest. That is according to their pakva or fitness. But to some, nothing except the
vichara marga [the path of enquiry] will appeal. They will ask, 'You want me to know or to see this or that. But who is the knower, the seer?' Whatever other method may be chosen, there will be always a doer. That cannot be escaped. One must find out who the doer is. Till then, the sadhana cannot be ended. So eventually, all must come to find out 'Who am I?'" - Be as you are, pg 76
Ramana never pushed people to give up other practices, and would even help people to do other practices that were important to them. However, it is clear that Ramana felt the finale of all practices would be the enquiry and that this was inevitable.
From Be As You Are:
Question: When I do this and cling to my self, that is, the 'I'-thought, other thoughts come and go, but I say to myself 'Who am I?' and there is no answer forthcoming. To be in this condition is the practice. Is it so?
Answer: This is a mistake that people often make. What happens when you make a serious quest for the Self is that the 'I'-thought disappears and something else from the depths takes hold of you and that is not the 'I' which commenced the quest.
Question: Yes. How to do it [Self-enquiry]?
Answer: Think 'I,I', and hold to that one thought to the exclusion of all others.