Sooner or later Amazon has to start paying dividends...
No, they don't. They just need to keep their share price afloat. Dividends are not a requirement of any company, and there are plenty out there that don't pay them.
Well, sure, in one scenario a company never turns a profit. In fact, many sole proprietorships and other small companies do just this, they just "zero out" by paying the principals the remaining profit as a bonus each year.
If, on the other hand, a company makes a profit, that profit must either be reinvested in the company or returned to shareholders in the form of dividends or stock buybacks. That extra money has to go somewhere, if only into the company's bank account (or embezzled by the CFO to pay for blow, but I digress). In Amazon's case, it's being spent expanding the company into new market areas. That process of reinvesting profits, however, is necessarily bounded -- they cannot become infinitely rich* or control infinitely many market sectors. So one of two things need to happen eventually: either they stop making a profit or else they start returning the profit to the shareholders. Any other state is transient -- although if they go through a number of profit/loss/profit/loss cycles, I suppose it could just wash out for a very long time.
So yeah, dividends are not a requirement. But they are a logical outcome of long term profitability because the opportunities for expansion are finite.