Try this
new_df = pd.merge(A_df, B_df, how='left', left_on=['A_c1','c2'], right_on = ['B_c1','c2'])
https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.merge.html
left_on : label or list, or array-like Field names to join on in left DataFrame. Can be a vector or list of vectors of the length of the DataFrame to use a particular vector as the join key instead of columns
right_on : label or list, or array-like Field names to join on in right DataFrame or vector/list of vectors per left_on docs