Are any two complexity classes equipped with an oracle to solve the halting problem equivalent?


Equip any complexity class $ C$ and $ B$ (to be more specific: any complexity class that contains only standard decidable problems) with an oracle $ O$ to solve the halting problem. Is $ C^O = B^O$ for any $ B$ and $ C$ that only contain problems decidable by a normal TM (meaning a TM with no access to an oracle (only the empty oracle))?