Mr Abbott's visit has attracted a far bigger media contingent from Australia than would be the case if all was normal back home.
Ms Bishop did not rule out putting her name forward for the leadership when questioned at a press conference in Auckland with Foreign Minister Murray McCully.
It was all hypothetical and speculation, she said.
The Liberal caucus voted on February 9 against putting the leadership to the vote but Mr Abbott's standing has been affected.
His controversial chief of staff, Peta Credlin, who is often blamed for Mr Abbott's internal party woes, has not come to New Zealand.
Mr Abbott and Mr Key laid a wreath at the Auckland War Memorial yesterday before attending the forum inside.
This morning they are holding formal talks at Government House where they are bound to discuss the impending joint training mission to Camp Taji near Baghdad.
They are in the awkward position of knowing it will occur but with Mr Abbott still not having announced an increase in its deployment from 600 to up to 300 more.
So it is something they will discuss in detail privately but keep under wraps until Mr Abbott believes the time is right to officially announce it.
Mr Key explained yesterday why he favoured a joint New Zealand-Australia mission as opposed to a more fully integrated Anzac-badged mission.
The reason, he said, was that he believed Australia would stay for longer and he intended that the New Zealand deployment - of 143 Defence Force personnel - would last for only two years.
"In the end, I think the symbolism of us working together is in the spirit of an Anzac force but it is not a badged Anzac force."
In his speech, Mr Abbott said that the performance of the New Zealand economy had made many Australians sit up and take notice. "Any sense of superiority felt on one side of the Tasman has been dispelled."
He said Australia and New Zealand were like siblings and the problem with siblings was that they could be taken for granted.
He suggested that New Zealand and Australia could work together better in the trade arena, in particular to pursue a free trade agreement with the European Union.
He expressed some annoyance that in previous free trade negotiations with Asean where transtasman co-operation was meant to have occurred, things ground to a halt as soon as New Zealand got what it wanted.