Revealing mistake: During the sequence with a crowd in front of the Gazette looking at the dummy on the stairs, there is a shot with the camera panning up (right after Kate arrives). Look in the crowd on the left side; a kid is following the movement of the camera on the crane instead of looking at the 'crime scene'. (00:11:50)
Continuity mistake: The quantity of blood behind Batman's dummy changes; in the overhead shoot of the crowd it runs horizontally along the step for an area 3-4 inches wide, while in the subsequent close-up there's no such trim. (00:12:00)
Continuity mistake: At the university, Sophie ends up chasing away anyone interested in Mary. She motions her to go and in a close-up says "Kinda looking for my future husband here." Behind her, a woman with a bike is walking right to left. In the next shot, the woman is walking starting from a point much further back than before, and even more noticeably, two girls calmly make their way halfway through the stairs and sit down...but they were already sitting there in that previous shot. (00:12:50)
Plot hole: Kate in the Batcave says with absolute certainty "Alice doesn't know that Bruce Wayne is Batman. Whoever stole the gun, knew that Bruce had to have the suit to test it on." Both statements are leaps in logic with no foundation. Alice knows that Kate is Batwoman, and that Bruce's office is her center of operation (she even shines a bat-signal there!). It would be perfectly logical to assume that she made the connection. The testing bit is simply a non-sequitur; plenty of weapons can be made and tested on armor which fits the specifications, real or inferred, of a particular target, without being in possession of the item as a whole. (00:16:00)
Continuity mistake: After some awkwardness in the elevator, Kate arrives at the party. Walks to the left and she comes across a blonde wearing blue pants and top. The woman holds her drink in her left hand, which becomes the right in the reverse shot. (00:20:30)
Factual error: The villain drops the elevator full of waiters and catering staff; the display gives him no indication of whoever is in each elevator, but we can maybe assume it was a staff-only elevator and he knew it (it surely had no markings to indicate it). When Kate reaches the ground floor, everyone inside that elevator is lying unconscious in the corridor. I could understand if the survivors crawled there, but everyone is simply KO meters away from the elevator as if the impact happened there and not in the elevator shaft with a devastating buildup that had to be vertical and contained for obvious reasons. (00:29:30)
Continuity mistake: In the elevator, Kate's dad is looking at the electric wires with a flashlight. His wife says "How can this city trust us to protect it", and he lowers the torch. Next shot, he has it still pointed at the circuitry, and he lowers it only a second later. (00:31:20)
Continuity mistake: In the lobby, Mary advances towards an injured woman she diagnoses from meters away. Seen from the front, she is holding her phone in the right hand. In the back view though, she has it in her left hand. The phone switches hand again when she commands the paramedics to go check her again. (00:32:45)
Continuity mistake: In her rooftop fight with Tommy, Batwoman opens with a batarang, then sends him face down kicking him from behind. They wrestle some, and the railgun goes off once. In a wide shot when she punches him in the midsection a few times while he is trapped in an armlock, you can see the gun on the ground. He had it in hand in the previous shot and is still in his possession later. (00:35:00)
Factual error: The villain triggers the detonation that causes the elevator with Kate's dad and stepmom to start falling. After an initial hiccup, the cabin begins a freefall. Only after this real fall begins and the elevator already fell a few floors down, Batwoman lands on another elevator. Pauses. Shoots her rope to the top of the shaft. Hooks it. Pauses. Shoots from Chekhov's gun the double edged harpoon, which effortlessly sticks to the top of the elevator cabin and halts the ruinous fall with a smooth slowdown and no oscillation. All of that happening because somehow the elevator in all those seconds is just a handful of floors lower, even if it was plummeting at breakneck speed from the very first shot. (00:35:40)
Continuity mistake: When Alice quips "And I am the crazy one...!" at the sight of Batwoman's wig, her hand is on the breach, then suddenly between Kate's hands. (00:36:40)
Continuity mistake: Tommy makes his unceremonious exit, and Kate spots Reagan. During the sequence with her, the same couple (she wears a reddish dress and has black hair in a ponytail) appears by the stretcher and then a few meters behind, even further behind when the camera is on Sophie reacting to her "There you are." (00:39:10)
Plot hole: The demented villain is not keeping tabs on the elevators! The rescue teams can move freely around the tower, the elevator doors can be pried open with ease like Sophie and her husband do, so his threat is completely empty and ineffective, somewhat surpassed in idiocy only by Batwoman's response, who during the ultimatum gets back home and keeps busy spraypanting the suit and finding a wig for her date with the crazy guy at the top of the hour rather than taking 10 minutes or so to free the people trapped in the 7 elevators first, unopposed as she is, and go challenge the idiot later when he has no more hostages. It shoud also be noted that the villain made the "hostage" situation and the "one hour" ultimatum known only to Kate! The police and the Crows have no reason at all not to intervene with full force to check out who the crazy bomber guy is, but the police does not swarm the building and nobody finds odd to see a madman on top of the building under terrorist attack.
Character mistake: Batwoman does everything she can to rescue the elevator falling down after the villain detonates the bomb, but does nothing for the second one Tommy blows up to make her lose her footing when she is vulnerable. There is no message from Luke telling her anything, and she is not shown having a HUD with thermal readings or something like that, so there is simply no reason why she would risk everything to stop one elevator and completely ignore the fate of the next one which could be full of civilians as well.
Who Are You? - S1-E4
Other mistake: Magpie is surprised by the guard at the beginning of the episode; the guard is running in the room, hand going for the gun. After a cut to her with close-up on her elaborate fetching of the bombs with her awkward nails, the man is standing still, hand down his leg, only then going for the weapon. Two bombs fall at his feet, one per foot, touching the feet directly. Obviously he just looks at what they are instead of jumping back, and they explode in a single mushroom cloud CGI explosion, to his left. (00:01:10)
Who Are You? - S1-E4
Continuity mistake: After the title screen, Sophie is reviewing Batwoman footage. First shot and two agents, one with a hat and one with blonde hair in a bun, walk behind her, right to left, met by another agent looking at something in her hands. Cut to the screen Sophie is looking at, and a few seconds later the same extras are doing their exact same walking routine starting from the exact spot. (00:01:25)
Who Are You? - S1-E4
Continuity mistake: In bed with the barista at the beginning of the episode, Kate puts her hand on her back. She says "I heard she's doing Batman" and Ruby Rose's hand is on her bare skin; reverse angle and you have the hand on the sheet that suddenly covers her. Next shot and the hand is on her neck, with hair rearranged accordingly. (00:01:50)
Who Are You? - S1-E4
Continuity mistake: Kate grabs her phone bedside, reading Luke Fox's message (complete with apropos bat-emoji). She puts it away and says "Sorry" to her fling, immediately placing the phone hand down her back, with the right hand being on her shoulder instead. New angle, and the hands switched position (left on shoulder, right hand at the middle of the back). After a few seconds of terrible lying, Reagan says "That sounds...really serious..." and you can also see a lot of hair moved in front of her shoulder. When they begin snogging before she goes there is more switching hand position business between cuts too, with the left hand being behind and in front. (00:02:30)
Who Are You? - S1-E4
Continuity mistake: In Bruce's office, the museum curator turns around to ask her assistant "Margot, is that the last lens change?" She initiates that line holding the handbag with her right hand, but in the reverse shot she is using that hand to point at her, since the purse instantly went under an armpit. (00:04:10)
Who Are You? - S1-E4
Continuity mistake: Somehow, Alice and a couple goons in conspicuous attire are just waiting by Catherine's car in the cemetery without her noticing. When she says "You know the thing I love about bad guys?" Elizabeth Anweis' hand is not in frame. Yet in the other shots you can see (use the jacket lapel as reference) that the hand, raised above the elbow level, should be visible. (00:06:20)