Using different passports for different countries?
 

My daughter is a US citizen. She is also an Indonesian citizen (as is my wife - though my wife is a US LPR). So when we travel to Indonesia and then Australia, we would like to use our daughter's Indonesian passport for her entry into Indonesia (thereby no need for Indonesian visa); when we go from CGK to SYD, we would like to use her US passport (thereby no need for Aussie visa). And of course, we will use her US passport when returning to US.

Any drawbacks to this approach? Btw, my daughter is all of 2 yrs old - no security risk unless a soiled diaper is considered a WMD.

Immigration officer may ask which country you just arrived from, and then couldn't find the exit stamp for it on the passport.

Its ok. If they ask, they show your other passport.

I have an ASEAN passport too in addition to my Canadian one. And it is handy travelling around the ASEAN regions that require $ for their visas.

Could the officer think that you're trying to hide something by using different passports?

Also, dual or multiple citizenships are not tolerated in many countries, so that may lead to complications if they find out.

No problem whatsoever. The US allows dual citizenship and the Indonesians allow it for children up to the age of 18. See http://www.indonesiamatters.com/830/dual-nationality/

Use her Indonesian passport to enter and leave Indonesia, and her US passport to enter and leave the US and Australia. Show her Indonesian passport at check-in when departing for Indonesia, and her US passport at check-in when departing for Australia and the US.

Millions of people have two (or three, or four) passports and travel without any problems on the passport of their choice.

As Guy Betsy said, you can just show them the other passport. I can personally say that I regularly carry two passports and have never been asked why I didn't have a stamp/visa for any country other than the one I am entering.

Not only can you do as many posters have suggested and the OP inquired about, it is the rule for most countries that you must enter/leave that country using the passport issued by that country. So for example, the OP's daughter must use the US passport to enter/leave the US, and perhaps Indonesia has a similar rule requing that as an Indonesian citizen, she enter/leave Indonesia on that country's passport.

Since of course you'll always have both...and both countries allow dual citizenship, shouldn't be a problem.