In the fencing bout on the rooftop, Hamlet and Laertes are dressed in modern foil fencing gear (with electric vests) but use épées instead of foils.
When we first see Claudius speaking, his mouth is obviously saying something other than what we hear. It lasts for about five seconds.
In the scene when Polonius gives parting advice to Laertes, the reflection of the microphone boom can be seen in the chromium décor of the room.
In Gertrude's bedroom, just after Hamlet kills Polonius, a boom mic is reflected in the windows.
In his soliloquy, Hamlet says "The undiscovered country to whose bourn /No traveler returns." Shakespeare wrote, "The undiscovered country *from* whose bourn /No traveler returns," i.e. no one comes back from the next life to tell us what it's like.