I have a dataframe, Date index type is Timestamp, Time column is datetime.Time:
Time Value
Date
2004-05-01 0:15 3.58507
20
You can first convert column Time to_timedelta, then add to index, drop column Time and if necessary set index name:
df.Time = pd.to_timedelta(df.Time + ':00', unit='h')
df.index = df.index + df.Time
df = df.drop('Time', axis=1)
df.index.name = 'Date'
print (df)
Value
Date
2004-05-01 00:15:00 3.58507
2004-05-02 00:30:00 3.84625
If column Time is datetime.time for me works cast to string first (if necessary add :00):
df.Time = pd.to_timedelta(df.Time.astype(str), unit='h')
df.index = df.index + df.Time
df = df.drop('Time', axis=1)
df.index.name = 'Date'
print (df)
Value
Date
2004-05-01 00:15:00 3.58507
2004-05-02 00:30:00 3.84625