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"It’s a word name which is fine, but it’s not a particularly positive one, as it implies naughtiness and mischief and constant running away."

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The meaning of Chase doesn't seem this negative. To chase after something is to pursue it, which to me implies ambition, dedication, tenacity, and following a dream or a goal. Chase, in and of itself, seems fairly positive to me. I do appreciate the irony of chasing after Chase.