You can convert the column first Time to_timedelta, then add to index, dropcolumn Timeand set the index if necessary 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 the column Timeis equal datetime.time, for me the conversion to is performed first in string(add if necessary :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
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