I went into the film with the twist slightly ruined for me by an FT feature about the gender issues it has stirred up. Fincher's latest effort (and in fact, most of his work) probably doesn't deserve the amount of analysis applied to it, since neither Amy nor Nick's characters can sustain a prolonged investigation into what makes them tick. But let's give it a go:
"Amazing Amy" has (due to her unique childhood) picked up the ability to identify and manipulate the various socially-prescribed roles foisted onto women. She is a femme fatale in the vein of Sin City's Ava – only in Gone Girl she is allowed to get away with it.
Why does she stay with Nick? Self-interest certainly plays a part, but there's also her delight in watching Nick learn his own role of adorable doofus. She has converted him to be her playmate in their sterile sham of a marriage – she can now manipulate him forever, his insides twisting pleasingly as he performs her every wish. Why write stories as wish-fulfillment (like Amy's mother did) when you can write reality itself?
The film ends on a slightly different note – generalising Amy and Nick's relationship into a comment on the institution of marriage as a whole. It does so by underlining the impossibility of ever really knowing what your partner is thinking, and how much of what you observe of them is role-play and bad faith. Fincher twists every sinew trying to extract as much horror from Rosamund Pike's final blank expression as he can. Whether it works depends on how much belief you can suspend in the film's ramshackle plot, which has Amy reacting to as much as shaping events, and which gives very weak motives for Nick staying with her and playing along.