Michael, why would you even consider winding it down?
First I need to say that I'm not an attorney and can't offer legal advice. For legal issues (and this may be one) you should always consult an attorney.
If you're concerned about brand, then create a new brand. If your entity has debt and you want it to file for bankruptcy, then you'll still have a problem since you can't just convey an asset that you have in the corporation to another entity.
To wind down and then create a new corporation is time consuming and costly. As long as your corporation is sound and you have been following all of the rules that make it a corporation (such as annual meetings and much more) why not keep it?
One possible consideration though is: do you even need a C-corp? Are you planning on making your company into one that VCs will want to invest in and you may want to provide options to employees? If the answer is yes, then it might be a good idea to have it. If the answer is no, consider a much simpler LLC or S- Corp that could provide better tax benefits. Of course as mentioned before, check with legal and accounting experts to find out which type of entity is right for you.