Bunzmann became a FIDE Master in 1994, and then achieved the title of International Master in 1999. He was awarded the Grandmaster title in 2020. Some notable events in his chess career include obtaining the second place at the German Youth Champion U18 in 1994 and winning shared first place at Open International Bavarian Chess-Championship in 2001. He was also the joint winner at the Augsburg Chess Festival in 2003.
Bunzmann represented Germany in the European Team Chess Championship in 1999. He was also seen participating in several FIDE World Championships, such as those in 2000, 2001, and 2004.