I have recently found out that my ex husband is a cross dresser and considers himself to have 2 personalities.

All cases involving relocating to another state with your child are difficult, if the other parent challenges it.  His cross-dressing may be odd, but it does not necessarily make him a bad parent.  You would have to show why it is in your child's best interest to move to NC, even though this will take her away from her father.  It can be done and his dual-personality may very well be a factor, but it will not necessarily be the determinative factor unless it adversely affects your daughter.  The court will look at ALL of the factors under Virginia Code 20-124.3 in deciding which of you should have custody:  https://leg1.state.va.us/cgi-bin/legp504.exe?000+cod+20-124.3  This answer is given in accordance with the laws of Virginia and may not be applicable in any other state.  It should not be construed as legal advice, as that would require a more thorough analysis of all of the facts involved in a specific case.  If you need further information or assistance, please feel free to contact my office for a consultation. Case results depend upon a variety of factors unique to each case. Case results do not guarantee or predict a similar result in any future case.