How to find the index of a value by row in a dataframe in python and extract the value of the following column

I will just using wide_to_long reshape your df

s=pd.wide_to_long(df.reset_index(),['Date','Age'],i=['Last_age','index'],j='Drop')
s.loc[s.Age==s.index.get_level_values(0),'Date']
Out[199]: 
Last_age  index  Drop
47        0      2             None
45        1      1       07/01/2014
74        2      2       08/01/1979
Name: Date, dtype: object
df['Last_age_date']=s.loc[s.Age==s.index.get_level_values(0),'Date'].values
df
Out[201]: 
  Last_Name       Date0  Age0      ...       Age2  Last_age Last_age_date
0     Smith  01/01/1999    29      ...       47.0        47          None
1      None  01/06/1999    44      ...        NaN        45    07/01/2014
2     Brown  01/01/1979    21      ...       74.0        74    08/01/1979
[3 rows x 9 columns]