If I am starting up a business in one state but noticed the name is already taken in a neighboring state, should I look for a new name?

When forming a business you may select between three basic types – partnership, LLC, or corporation.  Since partnerships are not registered with the state you can select a name that may be the same name as another business as long as the consumer is not confused.  Thus if the businesses are far apart or perform different functions you may be able to keep the name.  However LLCs and corporations are different in that they are registered with the state and so the name must be unique for the entire state.  In addition if you want to expand your business into the neighboring state it must be unique there.  Here from your facts it seems that you are creating either a LLC or a corporation and that your name is unique in your state but not in the neighboring state.  If the business in the neighboring state did not register their name then you will be able to register your name there.