Your question seems to infer that the main point of view or method of narration focuses on the central character, Peyton Farquhar. In fact, if we analyse the story carefully, we can see that this is not the case, and that the point of view seems to constantly shift throughout the tale. Note how the first section of this great story is narrated from an objective third-person point of view as we are given an observer's perspective of what can be seen. The second section again uses a third-person point of view, but we see it is not as entirely objective as we are given more insight into the character of Peyton Farquhar, as the following quote reveals:
No service was too humble for him to perform in aid of the South, no adventure too perilous for him to undertake if consistent with the character of a civilian who was at heart a soldier...
Note how this quote tells us a lot about Farquhar and his loyalties and how he struggles against being a civilian when "in his heart" he wishes to be a soldier.
Lastly, the third section shifts suddenly to a kind of third person limited point of view, as we suddenly follow the story from the perspective of Peyton Farquhar, zooming in on his thoughts and feelings as he falls off the bridge and the rope breaks. Of course, this continues up until the end of the story, when we are given another shift in narrative back to third person omniscient and we are told the truth: that Peyton Farquhar has actually died and is swinging from Owl Creek Bridge. Such jumps and changes in the narrative seem to reinforce the profound disorientation that Peyton Farquhar himself experiences and also give the reader the chance to participate in his final flight of fancy before his death.