I have a Client/Customer that has two Contacts. One of the Contacts has changed jobs and moved to a different Company.

That Company is also a Client/Customer in my account. 

It seems that there is no way to move a Contact from one Client/Customer to another. 

There is no way to remove a Contact from a Customer at all other than Deleting the Contact altogether.

That seems very strange. Contacts should exist independent of Customers.