What does Cathy mean by “I am Heathcliff”?

Cathy means that Heathcliff is not a separate object of affection but part of her own identity — that separating from him is a kind of self-division. She says this while explaining why she will marry Edgar: Edgar is handsome and rich; Heathcliff is now degraded. The speech holds love and social calculation in the same breath, which is why it is so unsettling.

It is not a wedding vow. It is a metaphysical claim made on the eve of a conventional marriage, and Heathcliff overhears enough to feel betrayed without hearing the whole confession. The line explains both the intensity of their bond and the disaster of treating that bond as compatible with Thrushcross Grange. Read it beside her deathbed reunion and beside Heathcliff’s later refusal to live in a world without her.