Merge pandas DataFrame columns starting with the same letters

Use dictionary comprehension :

df = pd.DataFrame({i: pd.Series(x.to_numpy().ravel()) 
                      for i, x in df.groupby(lambda x: x[0], axis=1)})
print (df)
   a  b    c
0  1  5  9.0
1  3  7  0.0
2  2  6  NaN
3  4  8  NaN

I'd recommend melt, followed by pivot. To resolve duplicates, you'll need to pivot on a cumcounted column.

u = df.melt()
u['variable'] = u['variable'].str[0]  # extract the first letter
u.assign(count=u.groupby('variable').cumcount()).pivot('count', 'variable', 'value')

variable    a    b    c
count                  
0         1.0  5.0  9.0
1         2.0  6.0  0.0
2         3.0  7.0  NaN
3         4.0  8.0  NaN

This can be re-written as,

u = df.melt()
u['variable'] = [x[0] for x in u['variable']]
u.insert(0, 'count', u.groupby('variable').cumcount())

u.pivot(*u)

variable    a    b    c
count                  
0         1.0  5.0  9.0
1         2.0  6.0  0.0
2         3.0  7.0  NaN
3         4.0  8.0  NaN

---

If performance matters, here's an alternative with pd.concat:

from operator import itemgetter

pd.concat({
    k: pd.Series(g.values.ravel()) 
    for k, g in df.groupby(operator.itemgetter(0), axis=1)
}, axis=1)

   a  b    c
0  1  5  9.0
1  3  7  0.0
2  2  6  NaN
3  4  8  NaN