"So what is going on really? From whose perspective did Bruce see all this happening? From his dream, it seemed like there was a third version of Bruce that saw everything happening".
12 Monkeys studies the subjective nature of memories and their effect
on perceptions of reality. Examples of false memories include Cole's
recollection of the airport shooting, altered each time he has the
dream, and a "mentally divergent" man at the asylum who has false
memories.
The short answer is that it's left up to the viewer to interpret the truth of the reality, being either a story about:
- A boy named James Cole who was traumatized at an indecent at the airport to the point that as an adult living in a mental institution, is making up a Dystopian time travel story about a future where man was nearly wiped out by a virus...
- or a man named James Cole who lives in a future where, as a felon forced to live underground, is tasked by scientist with time traveling back to figure out whom and what caused the virus in the first place in order to acquire a cure, and in which he has memories or a premonition recurring from childhood of an indecent that has yet to happen to his adult self that comes to pass at the end of the film (aka a paradox).
As many answers point out, you can find "flaws" or poke holes in both interpretations to the story that might sway a viewer to pick one interpretation over the other. (I however lean towards time travel with some kind of lightly altering casualty loop with not enough evidence to prove dramatic change to what may be the original time line or the previous iteration.)
This is why this film would categorized as a psychological thriller, because it's been designed for the viewer to constantly contest the nature of the characters' reality, as they move through the film and/or viewers may be left with questions and ambiguity still by the film's end in NOT being able to determine the full truth.
In this case the film is also playing with what is known as Cassandra Complex, relating to a metaphor originating from a Greek Mythological figure that believes: some things could be known in advanced.
The term originates in Greek mythology. Cassandra was a daughter of
Priam, the King of Troy. Struck by her beauty, Apollo provided her
with the gift of prophecy, but when Cassandra refused Apollo's
romantic advances, he placed a curse ensuring that nobody would
believe her warnings. Cassandra was left with the knowledge of future
events, but could neither alter these events nor convince others of
the validity of her predictions.
The metaphor has been applied in a variety of contexts, such as
psychology, environmentalism, politics, science, cinema, the corporate
world, and philosophy, and has been in circulation since at least
1949, when French philosopher Gaston Bachelard coined the term
"Cassandra Complex" to refer to a belief that things could be known in
advance.
This relates to the film in both that Cole knew about the "dream" of seeing himself and Cassandra (character also called Cassy) at the airport, but never took it seriously or knew how to apply it to his situation (as it kept "altering" each time he "remembered" it), playing into the tragedy of his character's story and the irony in that the scientists believed that they couldn't change the future by altering the past, by trying to prevent it. It never seems to cross their mind. They're only interested in the past in hopes of finding a cure, when Cole's memory suggest, they might of been able to dramatically change the future.