I am arriving at Zurich airport from Hong Kong and then going on to Copenhagen with a 55 minute transfer time, flying Swissair.
Do I need to go through passport control at Zurich or only go through Security check?
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Because you are entering the Schengen area in Zurich, you will go through passport control here. But you will not go through customs.
As soon as you pass through passport controls in Zurich, you will be at the Skymetro. Take it over to your departure terminal (e.g. A/B), visit your airline's check in desk if you don't already have your boarding pass for Copenhagen, then go through security check and to your departure gate.
You will claim your checked baggage and pass through Customs in Copenhagen. You will not need to claim baggage in Zurich.
The other answer is correct: you clear immigration in Zurich.
However, the order listed is wrong. Follow the signs to the A/B gates, and you will:
Take the escalator down to the underground Skymetro train, which takes you to immigration.
Visit the transfer desk if necessary.
Go to the gate