When an order is set to Complete, it doesn't mean that the order has been Shipped or Paid. It simply means that it has been authorized to be moved forward in the fulfillment process.
After the order is changed from Processing to Complete, the following will happen.
Payment, Shipping, and Export Status become available
The Status bars for Payment, Shipping, and Export statuses will appear and allow you to set them independently.
Inventory deduction is triggered
Inventory will also be deducted at this point if it has been set up and the order is At-Once.