I am STRONGLY left eye dominant, and shoot right handed. I always have - compounds for 12-13 years or longer, now I bowhunt with a recurve.
What, exactly, is wrong with shooting with one eye closed?
http://en.wikipedia.org/wiki/Depth_perception
Depth perception is the visual ability to perceive the world in three dimensions
Depth perception arises from a variety of depth cues. These are typically classified into binocular cues that require input from both eyes and monocular cues that require the input from just one eye
Binocular vision is vision in which both eyes are used together. The word binocular comes from two Latin roots, bini for double, and oculus for eye.
[1] Having two eyes confers at least four advantages over having one. First, it gives a creature a spare eye in case one is damaged.
[2]it gives a wider field of view. For example, a human has a horizontal field of view of approximately 180 degrees with two eyes but only 150 degrees with one[citation needed].
[3]it gives binocular summation in which the ability to detect faint objects is enhanced.
[4]it can give stereopsis in which parallax provided by the two eyes' different positions on the head give precise depth perception
http://en.wikipedia.org/wiki/Ocular_dominance
Also, field of view for binocular vision is about 180 degrees - field of view for monocular vision? 120 degrees.
Closing an eye you see, results in the loss of a lot. You don't close an eye hitting a baseball, throwing a football, serving tennis, driving, pouring a cup of coffee, typing on a computer ........ in fact, its very very rare for any human to close an eye in an activity because simply, its not how we're designed to work.