Brutus, those shoots are commonly referred to as knees. You can remove them, but they'll probably return.

Bald Cypress have a beautiful natural growth pattern. The only reason I would prune would be to remove dead branches.

Here is a link that might be useful: bald cypress

Natal, nice to see you here. I posted this message on another board and someone responded by showing me photo's of some cypress that are planted on a gulf coarse and their knees are mowed down at all times and the trees are beautiful. I'm glad to see that because I would like to try to do something under the trees, but those things get in the way. LOL Me knocking mother nature, I know, shame on me. I love the trees though, they are so pretty. I've never had a cypress on my property before. I have one on the side/front of the property that was struck by lighting during Katrina and it broke in half. Now it's so ugly, poor thing has lost its' natural shape. I won't cut it down though. I wish I could give it a nicely shaped top, but I know that's impossible. So it's a half a cypress!! When we cleared the property to build we had to take down some beautiful trees, some cypress. It made us sick to have to do it. I actually had people ask if I could build my house around one. LOL Yea, I want a cypress running up through my living room floor. Doesn't everyone?