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Hutton was found by a Royal Commission of inquiry to have planted evidence used to convict Arthur Allan Thomas of the 1970 murders of Harvey and Jeanette Crewe.
Yesterday Mr Bush said the eulogy was raised with him during the selection interviews.
"My response was that the eulogy was given with the best of intentions, at a private funeral, to a grieving wife, children and grandchildren. But in hindsight, I can see how others interpreted that. There was offence caused to others. I understand that and I actually apologise for any offence that was caused by a Deputy Commissioner making those comments."
A police review of the case is underway into the Crewe murder case after a request by the Crewe's daughter, Rochelle Crewe.
Mr Bush has said he would not take any part in that review. Police Minister Anne Tolley said it was appropriate for him not to be involved. However, she said the eulogy was one incident in an otherwise strong career.
"I've told Mike Bush it was a line-call to give the eulogy, and it is something he regrets. But it is important to see it in the context of one incident in a highly impressive 35-year career serving the public.
''Quite simply, Mr Bush is one of our most experienced and successful officers, he has proven himself both at home and abroad, and I believe he will be an outstanding Commissioner."
Mr Bush, MNZM, is the current Deputy Commissioner Operations.
He has been appointed to his new role for a three-year term, commencing on April 3.
As Police Liaison Officer for South East Asia, Mr Bush was the first New Zealand official to reach Phuket following the devastating 2004 Boxing Day tsunami, and was awarded the MNZM for his work during the operation.
He was also responsible for planning and operations for the 2011 Rugby World Cup, and led the rollout of smartphones and tablets for frontline staff which had contributed to an additional half a million frontline crime prevention hours each year, Police Minister Anne Tolley said.
Mr Bush has been Deputy Commissioner Operations since April 2011, and through Prevention First and Policing Excellence has managed a change programme in police which has contributed to a 17.4 per cent drop in recorded crimes over the past three years.
He has held challenging roles in rural, provincial and urban areas, and as District Commander in Counties Manukau he pioneered the implementation of Neighbourhood Policing Teams, which have successfully been introduced across the country, Mrs Tolley said.
The Governor-General made the appointment of Mr Bush as Commissioner of Police on the recommendation of the Prime Minister.
"I believe Mike Bush will be an outstanding Commissioner of Police, and will build on the excellent work of the current Commissioner, Peter Marshall,'' Mrs Tolley said.
"Mr Bush has devoted 35 years of his life to serving the public with the New Zealand Police, and I congratulate him on his appointment.''
Mr Bush said he was honoured to be appointed as the next Commissioner of the New Zealand Police.
"Since joining the Police in 1978 as a young constable I am passionate about this job. I come to work every day knowing that I am part of an organisation that has a major impact upon people's lives,'' he said in a statement.
"NZ Police has an enviable international reputation and we are proud to hold the trust and confidence of the public.''
He also paid tribute to the current Commissioner of Police.
"Peter Marshall has championed change at Police in order to deliver better results to the Government and New Zealanders and to better support the victims of crime. Under his leadership, reported crime and road deaths have continued to fall and Police are spending more time focused on supporting victims and the community through a prevention-based approach.
"We are building a truly innovative and modern policing service and I am committed to continuing that journey and taking policing to the next level,'' Mr Bush said.
"My leadership priorities for the role will be delivering the results important to New Zealanders, being safe and feeling safe in our communities and putting victims at the heart of everything we do.
"I will continue the visible leadership role of the Commissioner of Police, both within Police and externally.''