A Sphinx (Greek: Σφίγξ) is a mythical creature with the head of a human & the body of a lion. In Greek tradition, it has a human head, the haunches of a lion, and sometimes the wings of a bird. It’s supposed to be treacherous and merciless - those who can’t answer its riddle suffer the fate of being killed and eaten by the ravenous monster. Its what happens in the myth of Oedipus. Unlike the female Greek Sphinx, the Egyptian Sphinx is usually shown as male. The Egyptian Sphinx was viewed as benevolent, but having a ferocious strength similar to the Greek version. Both were thought of as guardians, often flanking temple entrances. The oldest known sphinx was found near Gobekli Tepe at Nevali Çori in Turkey. It was dated to 9,500 BCE.
Auto-enucleation (oedipism) and other forms of serious self-inflicted eye injury are an extremely rare form of severe self-harm that usually results from mental illnesses involving acute psychosis. The name comes from Oedipus of Greek mythology, who gouged out his own eyes.
Oedipus gauged out his own eyes finally at the realization of his deeds, a last ditch effort at denial, at not having to see the reality around him. As he did so, he uttered the words.
Oedipus: I am agony.
Words that resonate with Kaneki well.
I’ve pointed out before Kaneki’s character has built in allusions to the Oedipus Myth, but this is as close to a direct reference as you can possibly get.
Kaneki’s reason for doing so even seems to match Oedipus. A horrible reality he can no longer face, at the end point of his tragedy now that the truth has been revealed and he failed to protect his friends he can only look away for it has become too much for him.
When Oedipus saw her, with a dreadful groan he took her body from the noose in which she hung, and then, 1510 when the poor woman was lying on the ground— what happened next was a horrific sight— from her clothes he ripped the golden brooches she wore as ornaments, raised them high, and drove them deep into his eyeballs,  crying as he did so: “You will no longer see all those atrocious things I suffered, the dreadful things I did! No. You have seen what you never should have looked upon, and what I wished to know you did not see. 1520 So now and for all future time be dark!”
The question is why make such an obvious Oedipus reference at this point in time, now that Kaneki’s memories have already been recovered and the truth revealed. It has something to do with the other character who occupied that flashback.
In the myth of Oedipus, Oedipus kills his father and marries his mother. What makes this particular plot thread compelling is that he didn’t want to do either did, but still committed both unknowingly. He was led on by fate rather than his own desires, but still had to suffer the consequences in the end. And here too, Kaneki is compulsed in his actions not by his own desires, but by fate.
Kaneki’s reason for his flash back then,is probably deceptively mentioned with another character from a few chapters ago.
Kaneki is remembering his one good memory of Arima, and it’s a way of expressing his reluctance for what he is about to commit himself to doing.
Before Oedipus became King, he had to kill his father Laius the current king. At this moment Kaneki is committing himself to fulfill the Oedipal prophecy that surrounds his character by killing the only significant father figure he’s had in his life. He does so with regret though, because it’s not as if he wants to. Rather, he’s compelled, by fate, by the culmination of all of his decisions up until this point, call it what you will.