Past tense of 'fish': to catch fish or search for something in water or in a disorganized pile of things.
From Old English 'fiscian,' derived from 'fisc' (fish). The verb has been used since medieval times, and over time it expanded metaphorically to mean searching for something.
When you 'fish for a compliment,' you're actually using a metaphor that's been in English for centuries—comparing the careful act of fishing to the way people subtly try to get others to say nice things about them.
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