Emergency services were called to the Sydenham address at about 3pm on Saturday after other residents found a 50-year-old man with critical injuries.
He died soon after.
The victim has been named in court documents as Daniel Hawkins.
A 28-year-old man appeared before Judge Stephen O'Driscoll via audio visual link in the Christchurch District Court this morning.
He did not engage with the judge when spoken to.
He has been granted interim name suppression.
Court documents show he is facing a raft of other charges including the theft of multiple cars and a cellphone, giving false information to police, grievous bodily harm using a weapon and impeding the breathing of another person by strangulation.
He was remanded in custody until his next appearance in the High Court at Christchurch on May 25.
The New Zealand Herald cannot publish any further details about the alleged killer.
His lawyer told the court he was "not particularly responsive" when she spoke to him last night and this morning.
She suspected there were "underlying issues".
Hawkins was facing active charges and was on electronic bail at the time of his death, police said in court this morning.
Police are still at the complex carrying out a scene examination.
The process is expected to take several more days.
A post-mortem examination of the victim is under way this morning.
Tributes have been posted on social media for the man.
Last night, police appealed for sightings of a grey Holden Calais with the registration LDE987.
They said the car belonged to the victim and was believed to have left the property at 11.15am on Saturday.
It was abandoned twenty-minutes later on Monsaraz Boulevard near Halswell Rd around 11.35am.
The Herald has contacted the Ōtautahi Community Housing Trust who own the complex.
The site - between Gasson St and Waltham Rd - was previously a large social housing complex called Brougham Village.
However, after the February 2011 quake, 89 homes were extensively damaged and had to be demolished.
It has been rebuilt and tenants have been moving in stages. Construction is still being completed on some of the units.
