Eye Contact is Important
I find that I cannot make a true connection during conversation without making direct eye contact at least once. It lets the person know and feel that you are actually committed to what is going on. This does not mean that you must stare into their eyes the entire time, that would be a little awkward for just about anybody. However if you just take the time to plug in once, it seems to last throughout the conversation. There is always the chance that something can be said to break this connection. However it can be reestablished with another glance into the eyes of your fellow human.
The eyes say so much, they are a world of their own. This is probably why they look so much like the pictures of planets we are shown. If they are this powerful, then I do understand why some may be afraid to look into them. However we must break through this fear, I promise once you do you will enjoy the release you receive.
There are also people who take advantage of what their eyes are capable of, do be aware of this. I know we all know of that one person with beautiful those eyes that can get whatever they desire just by looking into someone. These kinds of people do not always deserve eye contact, unless they are willing to make their self vulnerable in some sort of way as to not catch us off guard.
The main thing I want to get across with all of this is that conversation is not complete until we say hello eye to eye. We should take pride in whatever our eyes look like, and learn to appreciate the uniqueness in others as well. We have the ability to go so much deeper with what we say, and all it takes is a quick moment to soften the situation with one look.