"I will see my doctor in a bit."
This sentence really should be 见我的医生/to meet my doctor, or just 看医生/to see a doctor.
We don't normally insert 我的 into 看医生. 看我的医生 is not impossible though to stress this is my own business to see a doctor (a more advanced expression).
I will go my own way to see a doctor a bit later. That's not your business.
(Note: I modified this part 1 day later to remove a misleading generalization. Anyway, there is a problem to express in a quasi-possesive pattern when the relationship is originated from the action itself.)
Beware of different uses of 看:
- 我去动物园看猴子/I go to the zoo to see monkeys
- 我看小孩/I watch the children (take care of them)
- 我看他不会来/I think he will not come
- 我看人不会错/I never misjudge a person