Moser started to play junior hockey at EHC Biel as a six-year-old. He captained their Junior Elite A team during his last junior season in 2017–18 and made his National League (NL) debut that same year, appearing in two games (one assist) with EHC Biel. Moser started the 2018–19 season with Biel's NL team and played 43 games (seven assists) and 12 playoff games (two assists). At the conclusion of the 2019 playoffs, he agreed to his first professional contract with Biel, signing a two-year deal on 10 April 2019.[1] The contract contained an National Hockey League (NHL) out clause for the 2020–21 season.[2] On 13 October 2020, Moser agreed to an early three-year contract extension to remain with EHC Biel through the 2023–24 season.[3]
Following his selection in the second round of the 2021 NHL entry draft, on 14 August 2021, Moser was signed to a three-year, entry-level contract by the Arizona Coyotes of the NHL.[4] He began the 2021–22 season with Arizona's American Hockey League (AHL) affiliate, the Tucson Roadrunners, but after registering 5 goals and 12 points in 18 games, he was recalled by the Coyotes in December[5] after Jakob Chychrun was placed on injured reserve and made his NHL debut on 15 December, in a 3–2 loss to the New York Rangers.[6] Moser scored his first and second NHL goals in a 8–7 loss to the San Jose Sharks on 28 December.[7] He remained with the Coyotes for the rest of the season. The 2022–23 season saw Moser play in all 82 games and develop into a key part of Arizona's defence.[8]
Moser made his debut with Switzerland men's national team in February 2019.[11] He was later selected to play for Switzerland at the 2019 IIHF World Championship in Slovakia.[12] Moser sustained a wrist injury during the third game of the world championship against Austria, forcing him to sit out the remainder of the tournament.