How long are you planning to stay in Australia?
The immigration officer wants to know how many days, weeks, or months Suzuki will be staying in Australia. He uses present continuous+infinitive to ask about a future plan which has been decided but could change.
Other examples are:
When are you planning to go to Italy?
How long are you planning to stay in Japan?
Are you here on business, or for sightseeing?
Remember that when asking about the purpose or reason for a visit, we usually use the preposition "on" with "business", and "for" with "sightseeing".
Are you here on business, or for sightseeing?
Be careful! This question begins with "Are you ...", which is the start of a yes/no answer question. However, the officer uses "or", so Suzuki has to make a choice. If a question uses "or", you cannot answer with "yes" or "no".
Another example:
A: Would you like tea or coffee?
B: Coffee, please.