What is unique about the window Barrymore is looking out? Why did he choose that window?

The window is unique because it is the one window in the house that looks directly down onto the moor.

The western window through which he had stared so intently has, I noticed, one peculiarity above all other windows in the house—it commands the nearest outlook on to the moor. There is an opening between two trees which enables one from this point of view to look right down upon it, while from all the other windows it is only a distant glimpse which can be obtained. It follows, therefore, that Barrymore, since only this window would serve the purpose, must have been looking out for something or somebody upon the moor.