This feels like the beginning of an adventure fic - maybe the girl turns out to be his daughter, and they hunt down her mother's murderer together while dealing with family issues...

One thing I don't get is why she came, though - she tells him she hates him, but does she have any further plan? If she thinks he really killed her mother, isn't it dangerous to be alone with him? He has a gun...or maybe she doesn't really think it, she's just looking for someone to blame.

I just rewatched the Big Sleep, with Humphrey Bogart - this feels like the kind of opening scene he would have, playing the private detective ;)