Is it permissible to be naked and completely undressed in an open area?

Complete nudity when relieving oneself is not forbidden (haram), but it is disliked (makrooh) if there is no need for it, because it goes against modesty before Allah. Allah is more deserving that one feels modest before Him than before people. Therefore, it is preferable to uncover the 'awrah only to the extent necessary for relieving oneself.