I read this short fiction (not very short, probably novelette, maybe even short novella, rather than short story) about 20 years ago. In was in a collection, not sure it was in English, possibly in French translation. It definitely does not sound like a French original SF.
I don't remember much. The POV character had his brain damaged (maybe he almost drowned). So they try an experiemental drug on him to repair the damage. But they give him too much and he becomes super-intelligent. So much so that he wants more and more intelligence. He manages to steal more of this drug and flees from the hospital (or experimental medical facility) where he was first treated. The authorities try to catch him but he is much too clever for them, especially after taking the stolen doses of the drug.
At some point he realises that there is another super intelligent person. He tracks him and they are both convinced that only one such can survive, Earth is too small for two like them. So they engage in a titanic battle of minds. But the other one has known of the existence of the POV first, has found him and had managed to imprint some kind of virus in his eidetic memory (IIRC, just by passing along him wearing a tee-shirt with a bizarre image on it). At some point of the mind battle the virus is activated and the POV loses, and dies.
