The son of a Mecklenburg judge, Bauch studied art history at the University of Berlin under Adolph Goldschmidt and at the University of Munich under Heinrich Wölfflin.
He wrote his Ph.D. dissertation on the Rembrandt pupil, Jacob Adriaensz Backer, at the University of Freiburg under Hans Jantzen.
From 1924 to 1926 he was assistant of the famous Dutch Rembrandt scholar, Cornelis Hofstede de Groot in the Hague.
For some years he worked as a Privatdozent, teaching medieval and early modern art in Freiburg and Frankfurt am Main.
After joining the Nazi Party in 1933, he was appointed professor of art history at the University of Freiburg, where he retired in 1962.