Alfonzo Dominico was very popular with the inhabitants of the town of Pearl Bay, much more so than Bucket himself.
Alfonzo Dominico was found floating in the Pearl Bay swimming pool in the second season of SeaChange, and during the trial of an accused murderer, his death was the subject of an inquest into whether the dog was really the personality Bucket never had. Alfonzo Dominico Jones was killed by poison left in his water bowl. The intention was to kill Bucket, who often drank from his dog's bowl when he was drunk.
The Pearl Bay swimming pool was dedicated to Alfonzo Dominico Jones's memory. This was when we learned that his second middle name was Antonio. For Bucket, it was the name that no one said that was the most important.