Yes, you can create a separate Alterdesk account for every organization that you work for.
If you have entered your phone number when registering for both accounts, they will automatically be merged. If you have registered without entering a phone number, you can still merge your accounts at any given time.
When merged, you can easily switch back and forth between your company accounts by clicking on the name of the organization at the top left of the screen and choosing the organization of your choice from the dropdown menu that appears.