Am I eligible for dual citizenship having been in Oz since 1959 but travelling on a Kiwi passport?

Hi. I was born in New Zealand to Australian parents and the family returned to Australia in 1959. I've only ever worked, paid taxes and voted in Australia but for personal reasons have always travelled on a Kiwi passport. Now I'm wondering exactly what I am! And whether I should chase 'official' Australian citizenship or just apply for an Australian passport. Any thoughts/advice is appreciated. Many thanks.