This is meant as a video version of a stage productions, so it is generally limited to what you would see on a stage without film enhancements. The notable exception to that is the use of camera views that I believe was used to replace the effect the live production had achieved in placing a large mirror in front of the audience. That generally works well, particularly as reference to ubiquitous CCTV cameras that would be familiar to people around London (making their way in NYC now, as well).
Some reviewers didn't care for another camera effect: that of actors addressing it directly when speaking some of the lines, particularly Hamlet for some of his soliloquies. I remember having the same reaction many years ago when watching a BBC performance of The Merchant of Venice in a Shakespeare class in college. Then I found it distracting, though not so much here. I believe that the reason why we find it jarring on a screen is that the close-up makes it seem rather like breaking the frame of the actors within a play when they come face-to-face (as if on a video call) with us. When you would see it on the stage, it would be far less intrusive because of the physical distance between the performer and the audience.
David Tennant is engaging with some of the same mannerisms he brings to the the Doctor's character, though without the sideburns. I found some of the mad antics a bit much. (Admission of personal bias: I'm give to understatement rather than overstatement and do read Hamlet as an introverted intellectual.) They must have been the director's idea, as Ophelia did the same kind of thing.
I didn't really care much for Ophelia who didn't get to convey any emotion other than hysteria and madness. When calm, she was rather wooden. As for stripping down to her underwear for her madness, some pointed out that it was inconsistent with the description of her drowning, which is attributed to the weight of her water-logged clothes. That's a good point.
I can think of two possible defenses for the direction here, though I wouldn't have gone that route myself. One is that Ophelia's mad entrance is described as entering with her down. For Shakespeare's audience, that would have been seen as a state of undress, as a properly put together young lady would have her hair contained. To achieve that in modern terms, the hair alone wouldn't cut it. Two: in King Lear, when Edgar assumes madness, he strips off his clothes, so that typical mad behavior in Shakespeare.
I didn't like the way the Mousetrap scene was done -- overly crude and in-your-face kind of obvious. I suppose they thought otherwise the Shakespearian double entendres would just fly over the audience's head. There is also the bit of anachronism of male actors for female roles when the play is set in modern times, though I suppose that may be meant to contrast the more realistic contemporary look of the production we are watching with the artifice of the players' performance.
One thing that I liked in that scene was the transference for the cameras. Hamlet rebels against being caught by the cameras and then uses one himself in recording the play and catching Claudius' reaction.
The king's reaction is much calmer in this production than the way it comes across in other. He carries a lantern over to Hamlet and shows him with a shake of the head that he knows that Hamlet knows, and he's letting him know that he knows. That may be going just a bit far, given that Claudius resumes the pose of liking Hamlet for the fencing match, and why would Hamlet go along knowing that he knows that he knows?
What I liked in the last scene was a kind of closing the circle visual. The cut Hamlet put into his own hand after speaking with the ghost, remains a reminder for a while with a bandage. At the end he transfers the same cut -- also on the left hand -- to Claudius. That makes the actor's controversial shrug a bit more understandable, as he is already poisoned.
As we don't have the textual ending of Fortinbras, the play ends with Horatio's emotional farewell to his friend, who has died in his arms. That makes it a rather more intimate death than one that would get the royal honors that Fortinbras said he would bestow on the prince of Denmark.
Overall: I wouldn't go so far as some who claim this is the definitive Hamlet for the 21st century, but it's one that most people would probably enjoy watching. I suppose that each generation has to endure assumptions about how Hamlet is to be made accessible to them. At least this one doesn't open with a summation like Laurence Olivier's opening to his film version: "This is the story of a man who could not make up his mind."
Speaking of how the generations reinterpret Shakespeare's play, you can watch a sort of companion to the production video, Tennant's exploration of the play's enduring popularity also available online (note the video is just a tad out of sync).
The most striking thing for me in that video was seeing the rare first editions of Shakespeare housed in the British Library handled so casually -- no gloves. When I had to use books in the New York Public Library's Arents Tobacco Collection, I had to be more careful with an Oscar Wilde typescript. But I was a mere student at the time, and one of my Grad Center professors was a disciple of Stephen Greenblatt.
Back then New Historicism was still, well, new. Greenblatt's appearance in this video indicates that his approach is not quite mainstream. Shakespeare is subjected to biographical details inserted into understanding his plays with the Hamnet/ Hamlet connection. The name of Shakespeare's son who died in childhood was Hamnet, which does sound an awful lot like Hamlet, but is not quite the same. It's hard to hear the difference in the video which accepts the connection as a given. It's Greenblatt's argument in "The Death of Hamnet and the Making of Hamlet." Some of the argument is rather forced, but I won't get into all of that here.
