Müller was born in Basel, Switzerland, on 20 April 1927, to Irma (née Feigenbaum) and Paul Müller. His family immediately moved to Salzburg, Austria, where his father was studying music. Alex and his mother then moved to Dornach, near Basel, to the home of his grandparents. Then they moved to Lugano, in the Italian-speaking part of Switzerland, where he learned to speak Italian fluently. His mother died when he was 11.