Chickens live around 5-10 years. The Colonel was already fully grown when Cayde-6 found him at The Farm. It's been five years since D2 launched, and if we assume that five years have also passed in-game and in the lore, then he should not be alive still.
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Now I just have a mental image of someone replacing the chicken every time it dies without anyone noticing. Then just tells everyone its the same chicken.