I think you can use groupby and a dictionary comprehension here:
>>> df
Address ID
0 12 A Aa
1 66 C Bb
2 10 B Cc
3 10 B Dd
4 12 A Ee
5 12 A Ff
>>> {k: list(v) for k,v in df.groupby("Address")["ID"]}
{'66 C': ['Bb'], '12 A': ['Aa', 'Ee', 'Ff'], '10 B': ['Cc', 'Dd']}