Catherine's quote, "I am Heathcliff" in Wuthering Heights by Emily Bronte is used to define not only the relationship of the characters, but also the relationship's effect on Catherine. Catherine is making a claim that is less about Heathcliff and more about herself.
By claiming this, Catherine is saying that Heathcliff is ever-present in her thoughts and feelings. He is a part of her in that she has inadvertently given part of her own existence over to him because of the power her feelings toward him have over her. It is a testament to the strength and depth of their relationship.