He played junior hockey with the Prince Albert Mintos and then went to Michigan Tech University from 1960 to 1964. There he led his team to win the Frozen Four NCAA title in 1962. Bauman and Montreal teammate Charlie Hodge shared goaltending duties in the 1967 NHL All-Star game, combining to record the first--and still only--shutout in the history of the showcase event. It was one of only three games Bauman played with Montreal before being selected by the North Stars in the 1967 NHL Expansion Draft. He played parts of two seasons with the Stars, and then returned to Alberta to play for Calgary in senior leagues.