The midfielder arrived at the Allianz Arena in August for an undisclosed fee after winning one league title, the UEFA Champions League and two Copa del Rey crown with Real.
Alonso admitted it was tough to leave Madrid, but said the challenge of a new club in a new country was something he could not turn down.
"It was not a simple decision," he is quoted as saying to TZ. "It is never easy to leave Madrid because they are a top club, but I felt it was the right time.
"I had the opportunity to go to another top club in another league. Maybe it was the last chance to live a new experience.
"I wanted a fresh start in a new city. I felt it was best for my career."
Since joining Pep Guardiola's side the 32-year-old midfielder has started seven matches in all competitions, helping Bayern to six victories.
The Spain international claims that he did not expect to play such a prominent role at Bayern so quickly following his move, but is thrilled with his early form.
"I'm delighted," he added. "I've fitted into the team well and have a good relationship with the coach.
"I talk to him a lot because I know what he wants from me, so it's been easy to come into a winning club like Bayern.
"I was surprised to be so important from the start because I barely knew the team, but football is a universal language."
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