Born in Prague (former Czechoslovakia), Baumann was raised in Canada after his family moved there in 1969 following the Prague Spring. The family settled in Sudbury, Ontario, where, at the age of nine, Baumann became involved in competitive swimming, training at Laurentian University.
He was named Honorary Colonel of 412 VIP Transport Squadron of the Royal Canadian Air Force based in Ottawa in June 2011.
Alex Baumann was inducted into the Ontario Sports Hall of Fame in 1999.
By the time he was seventeen, Baumann owned 38 Canadian swimming records and the world record in the 200-metre individual medley. He accepted a swimming scholarship and entered Indiana University to train under its legendary coach James "Doc" Counsilman; however with chronic pain in his shoulder, he made the decision to return to Sudbury to continue physical therapy and training under the direction of his long-time coach, Dr. Jeno Tihanyi. Forced to sit out the 1982 World Aquatic Championships in Ecuador due to his injury, Baumann's training progressed enough that he was able to win gold in both 200 and 400-metre individual medley events at the 1982 Commonwealth Games in Brisbane, Australia, lowering his world record in the 200-metre event to 2:02.25 in the process. It was here that he met his future wife Australian swimmer Tracey Taggart. He also won the 400-metre individual medley event at the 1983 World University Games.
Prior to the 1984 Summer Olympics in Los Angeles, Baumann's father died of complications from diabetes and his brother, Roman, died by suicide. Baumann persevered through these tragedies, as well as his lingering tendinitis and shoulder injuries, to enter the Olympics as one of Canada's best Olympic hopes.
At the 1984 Olympics, Baumann was selected as Canada's flagbearer for the opening ceremonies. He won gold medals in the 400-metre individual medley, setting a world record time of 4:17.41, and the 200-metre race, lowering the world mark to 2:01.42. The 400-metre gold was Canada's first in swimming since 1912.
He was named Canada's male athlete of the year for 1984 and was made an Officer of the Order of Canada. He was also named as the Male World Swimmer of the Year by Swimming World magazine in the same year. In 1988, he was awarded the Order of Ontario.
Baumann continued his swimming career with three gold medals at the 1986 Commonwealth Games in Edinburgh in the two individual medley events, and swimming anchor on Canada's 4 × 100 m medley relay team. However, later that year, he won a silver and a bronze at the 1986 World Aquatics Championships in Madrid, Spain losing to a 19-year-old Hungarian swimmer, Tamás Darnyi.
Baumann shared a close bond with his coach of 14 years, Dr. Jeno Tihanyi. After the 1984 Olympics, the two authored a book titled Swimming with Alex Baumann: A Program for Competitive and Recreational Swimmers.
Subsequent to the 1987 Pacific Championships in Australia, Baumann retired from competitive swimming, initially working with Canadian broadcaster CBC for the 1988 Olympics, and also returning to Sudbury as a coach until 1991. Some speculated that Baumann was unable to find a coaching position due to his outspokenness. He criticized the Canadian Olympic Committee for dismissing Don Talbot and replacing him with Dave Johnson, as it was widely alleged that Talbot was fired for refusing to put the daughter of a Committee member on the national team. Baumann also criticized Swimming Canada for its subpar performances at the Commonwealth Games in Auckland in 1990.
Baumann moved to Australia to enter graduate studies at the University of Queensland before becoming manager of sport programs with the Queensland Academy of Sport between 1996 and 1997. He then held various positions with the Queensland Government before becoming CEO of Queensland Swimming in 1999.
After a poor showing by Canada at the 2000 Olympics in Sydney, Baumann felt he had more to offer Canadian sport, and expressed interest in taking on the vacant leadership position at the Canadian Olympic Committee; however, the COC felt he did not have the business background to handle the job. Ultimately, Baumann was not offered the Canadian Olympic Committee position. This caused some controversy and criticism in Canada as many Canadian athletes and citizens felt he was the best choice for the job. A number of foreign sports organizations felt the same way, and he was courted heavily by the English Institute of Sport and the Queensland Academy of Sport, ultimately accepting the position of executive director for the Queensland Academy of Sport in 2002.
In 2004, after a dismal showing by Canada at the 2004 Summer Olympics in Athens, in which they failed to medal and their best finish was sixth place, head coach Dave Johnson was fired. The COC offered Baumann the CEO or head coaching position of Swimming Canada (Karen Spierkel had resigned as CEO on 3 May 2003 after the controversy surrounding Jennifer Carroll waving a Quebec flag on the podium during the 2002 Commonwealth Games) but he turned both down, saying that he would remain with the Queensland Academy. Several commentators panned Baumann as hypocritical as he was publicly critical of the Canadian swim program for several years from 2000 to 2004, right up to the Athens Olympics. In response, Baumann toned down his criticism.
On 27 September 2006, the Canadian Olympic Committee announced that Baumann was hired to take over Canada's Road to Excellence initiative beginning in January 2007. Canada had achieved mediocre results at the 2004 Athens Olympics with a tally of only 12 medals in all events, which resulted in an overall 21st place national showing. These poor results prompted much criticism from Canadians. At that time, Baumann announced a goal for Team Canada: a top 16 finish in the 2008 Beijing Olympics and a top 12 finish at the 2012 London Games.
On 15 January 2012, High Performance Sport New Zealand announced the appointment of Baumann to the role of Chief Executive, effective 31 January 2012.
In March 2009, Baumann swam at the Ontario Masters Swim Championships. It was his first swim meet in 22 years. He continued where he left off, beating the world record in the 200 metre individual medley in the 45-49 age group category by more than 3.3 seconds with a time of 2:12.01.
Baumann is married to Tracey Taggart, an Australian swimmer he met during the Brisbane 1982 Commonwealth Games. They married on 30 April 1988, and have two children: Tabitha and Ashton Baumann.
In 2012, Baumann received the Queen Elizabeth II Diamond Jubilee Medal. On April 7, 2016 his son Ashton Baumann won the 200m breaststroke final at the Canadian Olympic Swimming Trials on April 7, finishing the race under the Olympic qualifying time, earning a spot on Canada's Olympic team.