Amon was born in Bulls, and attended Wanganui Collegiate School. He was the only child of wealthy sheep-owners Ngaio and Betty Amon. He learned to drive at the age of six, taught by a farm worker on the family farm. On leaving school, he persuaded his father to buy him an Austin A40 Special, which he entered in some minor local races and hillclimbs along with practice on the family farm.

After his retirement from F1, Amon dedicated himself to running the family farm in New Zealand's Manawatu District for many years. After retiring from farming, he lived in Taupo in New Zealand's North Island. In the early 1980s he became more well known in New Zealand from test-driving vehicles on the TV motoring series Motor Show and later consulted for Toyota New Zealand, tuning the 1984 Toyota Corolla and subsequent cars for sale there. He also appeared in TV commercials for the company, where much was made of the acclaim he won from Enzo Ferrari. Amon participated in the 2004 EnergyWise Rally where he won ahead of Brian Cowan. Amon drove a Toyota Prius for the event.  Amon was involved in the design of the upgraded Taupo Motorsport Park circuit, used for the New Zealand round of the 2006-07 A1 Grand Prix season in January 2007. At the New Zealand Festival of Motor Racing in 2011, Amon's life and career were honoured with a selection of his cars being driven and also used the event to raise funds for the Bruce McLaren trust. Amon was also honoured at the festival in January 2013.  Amon died in Rotorua Hospital on 3 August 2016, aged 73, of cancer. He was survived by his wife (they married in 1977) their three children and their grandchildren. One of his sons, James, is a qualified High Performance personal trainer. He trained Central Districts Stags cricket team, and was revealed to be Brendon Hartley's personal trainer.

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