The incident happened at about 17:30 on Friday 25 November on Princes Street and Regent Road.
A 27-year-old man was on a number 25 Lothian bus travelling east on Princes Street when another passenger started to cause a disturbance.
They both got off near the Apple store where the victim was assaulted.
He was then followed to Regent Road, opposite St Andrews House, where he was attacked again.
The attacker then ran off up the steps towards Calton Hill.
Officers believe the man pictured will be able to help them with their investigation.
Det Sgt Neil Spowart, of Police Scotland, said: "This was a vicious attack that caused considerable alarm to the victim as well as other members of the public.
"Given the central location of this incident and the time of day it occurred I'm hopeful that someone has information that can assist us.
"As part of this inquiry we are keen to identify the man pictured in the CCTV footage and he, or anyone who can assist us in identifying him, should contact police immediately."
A CCTV image of a man police want to trace in connection with a serious assault in Edinburgh city centre has been released.