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I believe the creature ends up killing Elizabeth out of revenge because Victor wouldn't make a female creature for the current creature. The creature wanted a female partner because he wanted to live like the family he saw in the cottage. He told Victor that he will take her far away from society and live alone with her, if he made him one.
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I think the monster show Frankenstein that the basis of life is having someone there who is everything you aren't; someone who matches and compliments you. The monster represents basic man and his need to feel loved or belonging.