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A strong case can be made for either answer, depending on the degree of self-knowledge that you believe Amory has attained and what the nature of that knowledge is. Either answer can be supported by evidence from preceding events.
If you choose a period, you are providing this book with a definitive ending, conveying that Amory truly believes that he knows himself and "that is all." This ending implies that he does not wish to know anything more, that he has completed some sort of quest; this is all he shall ever know. If we believe that people are fluid and always changing, then this degree of self-knowledge will not last long, and so Amory has not truly come to understand himself. He knows himself at one moment in time, but fails to recognize his capacity for change.
A dash leaves the book open ended, almost unfinished in a way, with both Amory and the reader left unresolved. The dash suggests either that Amory recognizes his state of flux and that any self-knowledge must continually adapt, or that he does wish to know something more; now this is all he knows, but he hopes to learn other things.
Certainly this book is not a quest novel in the traditional sense that stories of the Holy Grail or of escapes from prisons are novels about quests. Yet, the narrative does have the flavor of a quest. Amory is obviously striving toward something, but what? This question is answered most simply, and probably most satisfactorily, by himself. The fact that the novel ends with an exultant claim of self-knowledge bolsters this argument, but the evidence for this answer runs still deeper. From the very beginning, Amory has attempted to understand who he is and how his surroundings have influenced the "fundamental Amory." We witness throughout the novel Amory being influenced by one person or thing after another (Beatrice, Princeton, Rosalind) and then needing to leave them behind in order to get back to the fundamental Amory. The goal of the quest is for Amory to understand who he is, and then to come to peace with whatever he finds.
Dick Humbird is Amory's ideal of the upper class, and yet Dick is not snobbish nor does he utterly conform to rules. He does what he pleases, and whatever he does seems perfectly acceptable, if not admirable. Dick seems to make everything work, and is Amory's hero. The fact that Dick came from "new money," rather than the more socially respectable old money, suggests the meaningless of old distinctions in this new world. Grace and savvy is what counts, not money. With his death, Dick becomes "unaristocratic" in Amory's eyes. After this, Amory seems to forgo his own attempts at "success," finding it futile. In the death of his hero, Amory is reminded of the brevity of life (though not explicitly). This seems to signal to Amory that he must live his own life and not try to become something else, since everybody ends up as an unattractive, dead mass at some point. Dick's death drives Amory sharply back toward the "fundamental."
