Bob,
There’s no problem with you using your key on two different machines, provided they are used only by you. However, if you have two copies open simultaneously it’s impossible to tell it’s ‘you’ using both of them. What I’d suggest is simply closing one instance on the computer you’re moving “from” before opening the application on the machine you’re moving “to”.
Do I need to purchase another license to be able to have both open at the same time?
If you really need both copies open on two machines simultaneously you’ll either need another single user key or a multi-user key, or you could try persuading our subscriptions folks into providing you an additional key for some sort of unique requirement you have. A temporary shortterm key for migrating off one machine onto another wouldn’t likely be an issue, for example.