Pandas: add a column to a multiindex column dataframe

I would like to add a column to the second level of a multiindex column dataframe.

In [151]: df
Out[151]: 
first        bar                 baz           
second       one       two       one       two 
A       0.487880 -0.487661 -1.030176  0.100813 
B       0.267913  1.918923  0.132791  0.178503
C       1.550526 -0.312235 -1.177689 -0.081596 

The usual trick of direct assignment does not work:

In [152]: df['bar']['three'] = [0, 1, 2]

In [153]: df
Out[153]: 
first        bar                 baz           
second       one       two       one       two 
A       0.487880 -0.487661 -1.030176  0.100813
B       0.267913  1.918923  0.132791  0.178503
C       1.550526 -0.312235 -1.177689 -0.081596

How can I add the third row to under "bar"?

It's actually pretty simple (FWIW, I originally thought to do it your way):

df['bar', 'three'] = [0, 1, 2]
df = df.sort_index(axis=1)
print(df)

        bar                        baz          
        one       two  three       one       two
A -0.212901  0.503615      0 -1.660945  0.446778
B -0.803926 -0.417570      1 -0.336827  0.989343
C  3.400885 -0.214245      2  0.895745  1.011671

If we want to add a multi-level column:

Source DF:

In [221]: df
Out[221]:
first        bar                 baz
second       one       two       one       two
A      -1.089798  2.053026  0.470218  1.440740
B       0.488875  0.428836  1.413451 -0.683677
C      -0.243064 -0.069446 -0.911166  0.478370

Option 1: adding result of division: bar / baz as a new foo column

In [222]: df = df.join(df[['bar']].div(df['baz']).rename(columns={'bar':'foo'}))

In [223]: df
Out[223]:
first        bar                 baz                 foo
second       one       two       one       two       one       two
A      -1.089798  2.053026  0.470218  1.440740 -2.317647  1.424980
B       0.488875  0.428836  1.413451 -0.683677  0.345873 -0.627250
C      -0.243064 -0.069446 -0.911166  0.478370  0.266761 -0.145172

Option 2: adding multi-level column with three "sub-columns":

In [235]: df = df.join(pd.DataFrame(np.random.rand(3,3),
     ...:                           columns=pd.MultiIndex.from_product([['new'], ['one','two','three']]),
     ...:                             index=df.index))

In [236]: df
Out[236]:
first        bar                 baz                 new
second       one       two       one       two       one       two     three
A      -1.089798  2.053026  0.470218  1.440740  0.274291  0.636257  0.091048
B       0.488875  0.428836  1.413451 -0.683677  0.668157  0.456931  0.227568
C      -0.243064 -0.069446 -0.911166  0.478370  0.333824  0.363060  0.949672