I think that Meursault is most interesting in the time after the murder, where he feels the struggle between his two selves disappearing, with Meursault the past slowly disappearing s he becomes engulfed in the predicament that he is in. For example, the entirety of Chapter 2, Part 1, is Meurault talking about every little thing he does and how it makes him feel, however he does so in the past tense, implying reflection of himself. Later on in the book however you notice a change in the voice of Meursault's narration and he in fact says in Chapter 1, Part 2, that he "had lost the habit of analyzing myself", which had the been the majority of his self-being in the first half of the book. I think that this is very interesting in that Meursault acknowledges that he feels different now and that he isn't quite the same, yet he cannot really place a reason for why. The reason for this being that he has "lost the habit" of analyzing himself so he could not see a reason for his change.
I think that this is a key statement by Meursault that has fallen upon deaf ears, if only because the lawyer he says it to does not know him prior to the murder. Even to those "close" to him, such as Marie or Raymond, this statement still would not have the same affect that it does on the reader, who knows in full how Meursault sees the world. Marie, Raymond, the lawyer, and everybody else only sees the Meursault that he projects, a character he plays based on how he sees the situation unfolding or how he wants things to go. Meursault is in a way a reality alterer for doing this. In his "Meursault the past" state, he changes the course of people's action by changing his own for the desired outcome. Following the murder, we see this on a much lesser scale, such as when he is questioned and asked why he shot the Arab man four extra times, he is stone-faced and does not know how to respond to the question, showing signs that "Meursault the present" is the one in the situation and he cannot think outside of the now. Later though in the same chapter, Meursault enters this altered state and tried to manipulate the judge to shut him up but it does not work as it has for him in the past, showing that his duality is fusing or is unable to work together to the same effect that it once was.
Why does this matter though? Why does the dual nature of Meursault matter, and does it even exist or is he just flaky and inconsistent? What makes this matter is that Meursault is unable to even understand or narrate his own life, so why does he feel like he can alter people's course by trying to manipulate them to his way? What about Meursault is different that he can act this way and nobody calls him out for it? Is Meursault called the stranger because we don't know him or because he is stranger than others we see him interact with and is always somebody new, no matter what the situation? Why does he change in so many ways so easily?