If this happens, it'll be either due to (1) a significant change in how the President is elected and subsequent replacement of the existing party system,, maybe with a normal multi-party one, or (2) just the 2-party system changing like it has before - the polarity of the two current parties with regards to the White supremacist system changed in the 1960s, and the Republican party simply did not exist before 1854, with the Whigs as the Democrats' opponents.

One thing that will not happen is a non-Democratic, non-Republican President without either significant voting reform or a party realignment (or both). And if the party system realigns without changing the voting method, parties outside the major two, whatever they turn out to be, will still have no chance to win.