In OTL, the original setup in the USA was to run one presidential election: the winner would become President, obviously, and the runner-up would become his Vice-President. (This was clearly drawn up when Washingtonian optimism that the USA could avoid organised political parties was still considered realistic).
Later, it was changed to the current system, where a team of two people run for both the presidency and the vice-presidency simultaneously for each party.
However, there's a third possibility: what if separate elections were held for the presidency and vice-presidency? What would US politics be like if you could, for instance, vote for a Republican presidential candidate and a Democratic vice-presidential candidate? (Or whatever - the POD for this would be early enough that the political parties might be different).