Pandas melt on MultiIndex columns

In case of plain dataframe, this should work:

import pandas as pd


df = pd.DataFrame({
    'a': ['a1', 'a1', 'a2', 'a2',],
    'b': ['b1', 'b2', 'b2', 'b2',],
    '2018 jan': [0, 4, 8, 12],
    '2018 feb': [1, 5, 9, 13],
    '2019 jan': [2, 6, 10, 14],
    '2019 feb': [3, 7, 11, 15],    
})

df = df.melt(id_vars=['a', 'b'], var_name='date', value_name='value')
df['date'] = df['date'].str.split(' ')
df['year'] = df['date'].str[0]
df['month'] = df['date'].str[1]
df.drop(columns='date', inplace=True)

Output:

    a   b  value  year month
0   a1  b1      0  2018   jan
1   a1  b2      4  2018   jan
2   a2  b2      8  2018   jan
3   a2  b2     12  2018   jan
4   a1  b1      1  2018   feb
5   a1  b2      5  2018   feb
6   a2  b2      9  2018   feb
7   a2  b2     13  2018   feb
8   a1  b1      2  2019   jan
9   a1  b2      6  2019   jan
10  a2  b2     10  2019   jan
11  a2  b2     14  2019   jan
12  a1  b1      3  2019   feb
13  a1  b2      7  2019   feb
14  a2  b2     11  2019   feb
15  a2  b2     15  2019   feb

If you have some multi-index in columns as mentioned in comment, here you can convert it to plain dataframe:

df = pd.read_csv('file.csv', header=[0,1])
df.columns = [' '.join(col).strip() for col in df.columns.values]
df.rename(columns={'a Unnamed: 0_level_1': 'a', 'b Unnamed: 1_level_1': 'b'}, inplace=True)

@KOB my answer can fit generally any csv file with 2 rows header, where part of columns are only on first row, and part on both first and second row. Based on your question this code will put correctly all headers as requested. while reading csv and a MulitIndex dataframe created:

df_multiidx = pd.read_csv('two_levels_header_file.csv', header=[0,1])
id_vars = [idv for idv in df_multiidx.columns if 'Unnamed' in idv[1]]
value_vars = [valv for valv in df_multiidx.columns if 'Unnamed' not in valv[1]]
df_multiidx= df_multiidx.melt(id_vars=id_vars, value_vars=value_vars,var_name=['year','month'])
df_multiidx.rename(columns={col_ren:col_ren[0] for col_ren in id_vars})

Output:

    a   b   year    month   value
0   a1  b1  2018    jan 0
1   a1  b2  2018    jan 4
2   a2  b1  2018    jan 8
3   a2  b2  2018    jan 12
4   a1  b1  2018    feb 1
5   a1  b2  2018    feb 5
6   a2  b1  2018    feb 9
7   a2  b2  2018    feb 13
8   a1  b1  2019    jan 2
9   a1  b2  2019    jan 6
10  a2  b1  2019    jan 10
11  a2  b2  2019    jan 14
12  a1  b1  2019    feb 3
13  a1  b2  2019    feb 7
14  a2  b1  2019    feb 11
15  a2  b2  2019    feb 15