Gary Currie, 39, from Ballieston, was pronounced dead at the scene of the tragedy on Port Dundas Road in Glasgow city centre.
His colleague, a 34-year-old man, was taken to hospital suffering from severe head injuries.
Hospital staff today said his condition was “stable”.
It is believed both men have plunged more than 140feet when the truck-mounted lift snapped as they stood in its basket.
One worker, who did not wish to be named, said: “It seems the man fell from a cherry picker.
“Apparently it snapped and he hit some scaffolding as he fell.”
A joint investigation has been launched by Strathclyde police and The Health and Safety Executive (HSE).
A spokesman for the HSE said: “The HSE is investigating in conjunction with Strathclyde Police.”
City Centre Police have also set up an incident room.
Buchanan House is occupied by several tenants – including Network Rail, Transport Scotland and Glasgow Caledonian University – and is currently undergoing a revamp.
Anyone with information is asked to contact 0141 532 3036.