Cambridge definition of "ultimatum":
a threat in which a person or group of people are warned that if they do not do a particular thing, something unpleasant will happen to them. It is usually the last and most extreme in a series of actions taken to bring about a particular result.
Was it the threat he made to Conklin that if he didn't leave Bourne alone he would turn the fight around?
Which he does in the 3rd movie, he exposes Blackbriar and finds out who he is by the end.
The first four movie titles mirror the book titles verbatim. The movie is called The Bourne Ultimatum because the book has that title.
I have no proof to justify this other than to point out that it would be an extraordinary coincidence that the first three movies and the three books by Robert Ludlum happen to have the same titles when their plots are so fundamentally different. To be clear: the books’ plots are fundamentally and almost completely different from the movies’ plots. If that is so, then why would the titles match verbatim? My answer is perhaps the movies are acknowledging their inspiration from Robert Ludlum’s books of the same name. Please note that the fourth movie’s title, The Bourne Legacy, matches verbatim with Eric Van Lustbader’s first Bourne book. I have not read Lustbader’s books, so I cannot speak to their plots.
The word ultimatum fits with your comments. But, I would go further to say that a fair number of ultimatum’s were spoken throughout the movie series.
The movie's name is drawn from the novel of the same title by Robert Ludlum.
In the novel, the "ultimatum" is the final set of conditions concocted by the evil mastermind, Carlos the Jackal, to crown his life's work. They are to (1) destroy the KGB spy center that trained Carlos himself, and (2) kill Jason Bourne. These goal conditions are an ultimatum to himself, not to anyone else.
Ludlum could have alternatively entitled the work, "The Jackal's Bucket List."
