Translation:After one day at the airport, we began to complain.
"We began to complain after one day at the airport." should be accepted. It is identical to the given translation.
What about Present Perfect? "We have begun complaining after one day at the airport."
I don't think that would be right, since the Czech "začali jsme" fully puts the sentence in the past.
Here is my GUESS...
I believe začít" is a perfective verb, making it a "one and done" thing. In this sentence, the complaining STARTED once. A different verb (začínat?) would be used to indicate that the complaining started and is continuing, as would be the case with "have begun."
Perhaps a volunteer or more advanced learner will join the discussion to clarify if I'm wrong about this.
No, I don't think the sentence implies whether the complaining still goes on or not. The imperfective verb začínat would mean the starting was continuous, not the complaining.
But what it does imply is that the starting is already finished.
Thanks for the fix. My answer was misleading, now that I've read it again!
stěžovat uses SI, not SE. And there is already one SI from "začít si" so we do not duplicate it.