Julia Gillard also weighed in, saying she could "understand why people would want to go and live in New Zealand given the leadership of the current Prime Minister".
Australian Treasurer Scott Morrison will become Australia's new prime minister after winning a Liberal party leadership vote on Friday, ending an internecine battle that has scarred the conservative government ahead of an election due by May 2019.
Chinese foreign ministry spokesman Lu Kang congratulated Morrison on his new role and said Beijing would work with him to develop China-Australia ties "along the right track".
Australia is yet again in the midst of political instability, with Prime Minister Malcolm Turnbull widely expected to be forced from office in the coming days. The names would prove a majority of his government had abandoned him.
The prime minister said he was "optimistic and positive" about the future of Australia under the leadership of Scott Morrison.
Turnbull, who pledged to quit parliament after his near three-year reign came to an end, survived one attempt to oust him on Tuesday, but ministers then began defecting, throwing the government into crisis.
Members of Australia's conservative Liberal Party are meeting at Parliament House to choose a new prime minister.
Other favorites include Treasurer Scott Morrison and the only declared challenger Peter Dutton, a former Cabinet minister.
Cohen called state tax officials soon after subpoena
Mr Cohen's admission has thrown Mr Trump's presidency into crisis and raises questions about the President's own legal jeopardy. During the campaign, each had taken a six-figure payment that kept their claimed affairs with Trump out of the public eye.
Morrison will therefore need to find a way to meet Australia's commitment to reduce emissions by 26 percent from 2005 levels by 2030 under the Paris accord without stoking old divisions, a challenge Turnbull said would be hard.
Outgoing Australian Prime Minister Malcolm Turnbull waves while holding his granddaughter Alice during a final press conference before leaving Parliament in Canberra on Friday. "This will be a good thing for politics across the board, because the Liberal Party needs time for reflection and regeneration", said Chris Wallace, a research fellow at the Australian National University. A complication had been finding 43 lawmakers to sign a petition demanding a change of leader.
"We intend to be governing ... so I don't think anybody should be making any plans for any elections any time soon", Morrison said.
Turnbull told reporters after the vote he would leave parliament "not before too long".
Abbott led the Liberal-National coalition to a storming victory over Rudd's Labor in 2013.
Morrison and Bishop, who are both from Turnbull's moderate wing of the party, stood against Dutton in a bid to nullify his conservative alliance.
His departure from politics would spark a by-election for his Sydney seat, threatening the government's one-seat parliamentary majority. He also promised to reverse increases in electricity prices, which are a sore point in a country with huge deposits of coal and gas, and pledged to protect the universal health care system that is a source of pride for Australians.