In the moments before Peyton Farquhar is hanged, he "closed his eyes in order to fix his last thoughts upon his wife and children." It makes a great deal of sense that a person, in their final moments, would visualize the people they love best in the world and try to focus on them in order to make their last thoughts happy ones.
His mental journey, then, the journey he imagines as he is falling to his death, is to be reunited with his family (this journey constitutes the majority of Part III as Part II gives readers the background information we need to understand why he is being hanged). He imagines quite an epic trek: first he must swim through the rushing water and volley of bullets as the Union soldiers attempt to kill him again, then he must walk many miles in the dark through a forest. But "The thought of his wife and children urged him on." Finally, after having fallen asleep while walking, "He stands at the gate of his own home." The last thing he is aware of is his beautiful wife waiting for him "with a smile of ineffable joy, an attitude of matchless grace and dignity." Thus, Farquhar very successfully visualized his family so that they would be the last thing he thought of before he died. He didn't just visualize them but dreamed up an entire scenario in which he escaped death so that he could return to them.