You can take better advantage of the shape of most monitors by taking your images in portrait mode (with the camera on its side) - once you place the two images next to each other and scale the result it will fill more of the monitor.
If you want your image to look somewhat natural, then have the separation be relatively close to the normal separation between your eyes.
If you want a hyperstereo image, then you must be sure the two images are aligned. The best way to do that is to find a long, level, straight platform of some kind to set the camera on. I have used railings of various kinds with good success. Try to keep relatively close objects out of the image. The best images are of distant objects like mountain ranges and clouds. Cityscapes also make good subjects.
I think of hyperstereo images as monster-vision, picture a gigantic monster with eyeballs 4 meters apart and you get the idea.
If you want a cross-eyed image, then put the rightmost image on the left, and the leftmost image on the right (in other words, just swap them). For a wide-eyed stereo image, leave them in place.
This is one of my favorite stereo images. It powerfully demonstrates the human brain's three-dimensional modeling capability: even though the shadows contain little or no detail, when viewed in stereo the scene jumps out in stunning depth.
I'm very fond of GIMP's Adaptive Contrast Enhancement plugin, it's very good at enhancing the separation detail in stereo images.