You can rename
the index level to something more obvious:
df1 = pd.DataFrame([[1,2,3,4], [1,2,3,4]])
df2 = pd.DataFrame([[1,2,5,4], [5,2,3,1]])
df1.compare(df2, align_axis=0).rename(index={'self': 'left', 'other': 'right'}, level=-1)
0 2 3
0 left NaN 3.0 NaN
right NaN 5.0 NaN
1 left 1.0 NaN 4.0
right 5.0 NaN 1.0
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