As soon as he saw the hulking, bone-white carcass of an ocean sunfish sloshing in the waves offshore of the Azores archipelago, José Nuno Gomes-Pereira suspected the animal might be a record-breaker.

The type of eel-like fish, which recently went on display in Canada, is a fascinating denizen of the deep sea. The museum’s specimen was caught ten years ago in Queen Charlotte Sound (map), off the ...