Spoiler Alert!
For much of the film we know this character as Madeleine Elster, but her name is really Judy Barton.
It turns out that Judy has been having an affair with Madeleine's husband Gavin – an old acquaintance of James Stewart's character, retired San Francisco detective John "Scottie" Ferguson. Gavin has murdered Madeleine, and Judy impersonates Madeleine in an attempt to convince Scottie (and thus the world at large) that she is going mad. Judy and Gavin eventually convince Scottie that he has witnessed Madeleine's suicide.
Needless to say, Scottie gets to the bottom of all this in the end. He and Judy fall in love, but just as we begin to think they may live happily ever after, Judy meets a tragic end.
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