This behavior is normal and romantic relationships can develop between two people who spend a lot of time together in any situation may find they have feelings for each other and the workplace is no exception. How this can influence things at work depends on if the relationship works out or not.
If you are considering dating someone at work check out the HR policy on it first, you may find you can get into some trouble if you date someone within the company. Otherwise do what you can to keep things under wraps while you are at work for two reasons.
No one needs to know your business and if there is a policy against co-workers dating then you will stay out of trouble. A third reason is that you both will not become fodder for the rumor mill.
When the dynamic changes between two people at work and they decide to commit to each other in a serious manner care must be taken to keep productivity at the same level or higher otherwise the boss will suspect something is going on and she/he may question you.
Do what you can to stay as productive as you were before the relationship started and you will not have any problems. Do not spend all of your time sending e-mails or calling the other person on the phone. There is a time and a place for all of that and at work is not that time or place.
Make plans to meet for lunch before you get to work. Then at lunch make plans for after work. Try your best to keep your relationship out of the workplace. You may be so excited and feel like telling some of your closer co-workers about your new love but refrain from doing so. Word will get out if you do this and this is when the trouble we talked about earlier can start.
No one needs to lose their job over falling in love especially in this economy where jobs are hard to come by. Do everything you can to keep the one you've got.
If the newness wears off and the relationship starts to go sour do your best to keep the breaking up out of the workplace as well. Do not let your productivity waver in this situation either. Deal with the break up on your own time.
Romantic relationships can be tough at any time but between two co-workers it needs special attention. If things do go sour and you work in the same department then one of you could ask for a transfer to another department to stay out of trouble. You know what they say, "Out of sight, out of mind".