At the beginning of Doctor Strange, Strange is operating on a patient just getting ready to take the bullet out of their brain while the other doctor watches on. Strange asks ... no, tells the other doctor something about his watch. The other doctor then covers the watch up. What was going on with the watch which caused Strange to behave like this? (Other than him being strange ... er ... eccentric, that is ...)
In that scene, Strange was trying to focus (might be showing off as per West) to take out the bullet from the patient's head without using scan reports (directly using micro glaces). But he was disturbed by West's watch noise and he says "your watch is disturbing me". You can hear the watch's ticking sound in that scene. So West put his hand on his watch and then Strange removes the bullet from the head.
But when they got out, Christine asks him that there was no need to make fun of West. So you can make it that Strange was just humiliating West on that scene.
I think he told him to cover his watch as a reference to when West prematurely called the time of death. Kind of like a reference to not needing to know the current time when he extracted the bullet because the patient was going to make it.
Time is a major part of this comic. Strange (at this particular time in the plot) is still governed by time's constraints. His understanding is that time will wait for no one, bend for nothing, travel in no way but forward. This is terrifying for someone who has a near phobia of failure. That ticking clock got to him on another level since it brought into his narcissistic perception a loop of consequences that could lead to a tarnished reputation, or an ego stripping reality that he is not in control.
Later, in the alley in Katmandu, where the thugs stole his watch was significant in the respect that Mordo got the watch back, however it was broken. Symbolizing a new paradigm of chaos for someone who avoided giving up control at all costs.
My last opinion is pertaining to the ending of the film. After all the time manipulation to overthrow the evil one and prevent the Earth from being taking into the dark dimension... Dr. Strange is shown strapping on that same broken watch, even though he has that amazing spinning drawer of working designer watches. He seemingly embraces the fact that most everything is out of his control. Time is relative for each and every one of us. However, time no longer is tied to his success or failures. No longer dictates his path, or controls his actions. As the Ancient one said, "death is what gives life meaning." She no longer held onto the same meaning since she was breaking forbidden rules to empower herself with magic to grant life looong beyond her natural years. Breaking the natural law to protect herself (used to protect those around her as well)... Strange was breaking the natural law to save others in much the same way. (the librarian... the earth) Is this respect, he came into his own, and now understands that everything is not about him.. but aimed at a higher purpose.
I think we're missing the more subtle undertones as well. The issue with the watch is multiple. The movie is about time. The focus on the watch in the beginning cues to that. The insult to the other doctor is the fact that he failed on the patient originally, and called the "time of death" prematurely. The other side, meaning that now that Dr. Strange was working on the patient, he was going to have more time.
I thought it was at first that he was using a magnet to take the bullet out of the gentleman's brain. Then when Christine says the line about humiliating West so you know it is about West calling the time of the death of this poor gentleman.
The other doctor was wearing a cheap watch. Strange was obviously a watch collector and his JAEGER LECOULTRE that was broken during his accident was OBVIUOSLY a beloved time piece. The other doctor's watch simply wasn't up to Strange's standards thus he was making fun of him via his watch.