The first DLC for Borderlands 3, Mistxi's Heist of the Handsome Jackpot, introduces an array of old and new characters to the Heist. One particular character you might have questions about is Digby Vermouth, the resident musician who does sweet melodies on his saxophone.
You must have purchased the Moxxi's Heist of the Handsome Jackpot DLC to meet Digby Vermouth.
Some may recognize his name because of his song on the official Borderlands 3 soundtrack. He is credited with playing Supernova Dreamsicle. It wasn't from him, but you can check out his song below. It's a smooth, jazz-based song with some decent notes and a background flow. You might also have heart that Sir Hammerlock refers to himself as "kind of Digby Vermouth's 9th Concerto in D minor" when he explains his relationship with Wainwright.
You first meet Vermouth at a little alcove bar in the middle of the Spendopticon Vice District. You will need to do several quests for Vermouth, the first revolving around collecting ingredients to make him a drink to awaken his senses.
He explains how he's been stuck in the Handsome Jackpot since arriving on board, and Handsome Jack perished in Borderlands 2. Although he's stuck, he created jams for the whole resort, and it was a fun party for him. .
The next part of his quest allows you to locate his trusty saxophone from someone who stole it. You will continue to talk to Vermouth the entire time, learning about its background and Borderlands universe. You will also need to escort her to Foxxii for the third part of her quest.
Beyond seeing Vermouth in the Handsome Jackpot and playing a song with him, he doesn't show up or do much else. It's a funny little nod from the developers of the joke they got from the jazz musician from their video game soundtrack.