I couldn't miss this now, on all SE sites:
What are the duck powers? Was anyone able to make it do or say anything other than "Quack"?
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It is not exactly a feature but is a merely a legend (I don't have a better word). They basically want you to listen to your self twice and they believe that you will have your answer sometimes in the question itself. Go to Confused Retriever for more info.
According to wikipedia
A rubber duck in use by a developer to aid code review In software engineering, rubber duck debugging or rubber ducking is a method of debugging code. The name is a reference to a story in the book The Pragmatic Programmer in which a programmer would carry around a rubber duck and debug their code by forcing themselves to explain it, line-by-line, to the duck.
It does not matter if you have a mic or not. If you select "Yes" it will ask you to speak naturally but if you select "no" then this happens:
No matter if you say something out loud and clear it always gives the same output. It will ask you to speak louder and then do the silly "Quack" in the end.
Quack Overflow says,
Whenever you’re stuck, rubber ducking is a powerful method for solving even the most difficult problems. Insights are often found by simply describing the problem aloud.
You can then either select the option "I have another problem", which repeats the same thing over again, or you could select the "I hate this duck" option, which will display the message below and a button with the text Goodbye,duck.
The duck is sorry to hear that, but it understands.
But don't do this. It makes her cry :(
When does this duck appear?
I don't know. No matter how many times I hit refresh, go though questions or do anything it isn't coming back once I said "I hate this duck". :P
For really a technical point of view regarding this you can read this answer.
BUT again, you never know April 1.
How to get the duck back once you hated it?
According to @Simon's comment and this answer I found out that a cookie is being stored once you hated the duck. Deleting the cookie gets the duck back.
Some issues (problems?) get resolved by just being patient "a bit". Pretty sure this "duck" is related to april-fools indeed (as per one of the tags you also used in your question). So it'll be interesting to see what will happen to it in about a day from now (the bets are open ...).
And even though SE sites appear to always enforce that any user around the world is operating from the same time zon (i.e. UTC, e.g. for review queue stats, rep rankings, etc), it might be that in this case this special feature is an exception to it. As it seems to have been activated around the time that April 1st started in the very first real timezone (somewhere in the Pacific?).
Recently, I spotted a little "duck" at the bottom right of my screen:
I clicked on it, and it asked me if I had a microphone:
I clicked "yes", and it then started listening to me:
I then whispered it a very personal question, and it had a bit of trouble listening to me:
Eventually, it got through, and it responded with a loud...
And left me with some wise advice: