The assassins job was to kill Conklin. The very next scene is Abbott at the office giving orders, Meaning he had ordered that hit and was now in control.
Ephemeron.

As other posters said, the DVD alternate ending makes it clear that the assassin intended to kill Conklin. However, 12+ years ago, before any of the sequels, it seemed that Conklin's rumpled overcoat-in-silhouette was meant to be mistaken as Bourne in a rumpled-overcoat-in-silhouette.

Its built up for you to believe that he is going to kill Bourne, but pretty much what happened was the roof of Treadstone came crashing down, and Abbot knew Conklin had lost control and was a case of damage limitation.
Abbot knew Bourne was not dead (if you watch the DVD for the alternative intro and ending he actually offers Bourne the chance to come back to work for them).
Naturally it takes on a whole different meaning when we get to Supremacy.

Its kind of explained in the next one although not much but he definitely meant to kill Conklin