Cricket South Africa said on Monday that Boucher will leave his post after the October 16th-November 13th World Cup in Australia to pursue "other opportunities".
Former wicketkeeper Boucher took over at South Africa in 2019 and led them to 11 test wins, 12 victories in one-day internationals and 23 in T20 matches.
Boucher said it was an "honour and privilege" to be appointed head coach of Mumbai Indians.
"Their history and achievements as a franchise clearly put them up there as one of the most successful sporting franchises in all of world sport," he added.
"I look forward to the challenge and respect the need for results. It's a strong unit with great leadership and players. I look forward to adding value to this dynamic unit."
Boucher's appointment comes after Mahela Jayawardene stepped down as head coach to take on a bigger role at the franchise due to its increasing international presence.
Former India fast bowler Zaheer has been elevated to global head of cricket development from his previous position as director of cricket operations.
Led by India captain Rohit Sharma, Mumbai are one of the most successful teams in the IPL, winning the tournament five times, but they had a disappointing 2022 season where they finished bottom of the table.
Former England head coach Bayliss will join Punjab Kings for the next IPL season after the team did not renew Anil Kumble's contract. Under Kumble, the team failed to make the playoffs for three consecutive seasons.
Bayliss was England coach when they won the 50-overs World Cup in 2019 and also led the Kolkata Knight Riders to IPL titles in 2012 and 2014. Most recently, he was the head coach of Sunrisers Hyderabad in the 2020 and 2021 seasons.
"I'm honoured to be given the head coach's role with the Punjab Kings. A foundation franchise with an appetite for success. I look forward to working with a talented squad of players determined to compete for silverware," Bayliss said.
The next IPL season is scheduled to begin in late March 2023.
