Maybe I am a biased Aussie but I would go Melbourne for the lifestyle and climate.
Unless the Doha job is fantastic.
You have an advantage in that you already have an idea of what living in the Middle East is like. So you can make an educated decision. You probably already have a reasonable idea of Oz life (from TV and Aussies in UK etc).

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The superstition in which we were brought up never loses its power over us, even after we understand it.