American President and Vice-President Voted In Separately?

I recall reading once that someone thought the Office of Vice-President was undemocratic, since it was only the President who had a say in who it was. Not only that, but the President and Vice-President are voted in on the same ticket, which can in some cases turn people off.

So, what if the Vice-President was voted in separately from the President?

I for one am not exactly sure how this would work, but I suspect those of you who are more familiar with the American Election process would have some ideas.

For one, is this even possible, and if so, how might it work?
 
Many American states have separate ballot sections for Governor and Lt. Governor. The candidates usually run as a team, but they are picked independently using the normal convention/primary process and sometimes the election winners will be from different parties.
 
I would think this would require more defined duties for the VP than currently exist; if the Veep has nothing to do but break ties in the Senate and wait for the President to die, that's not a job any reasonable person would apply for, and you wouldn't like the sort of person who would apply for it, especially when a President really does die.

The concurrent portfolio thing is an excellent idea; I like the idea of giving them administrative power in the Senate but that probably warps the existing Constitution too badly. Say the office of VP and SecState were combined (as stipulated in the Act which establishes the State Department)? Thus the President picks his Sec's of Treasury, War and Navy, and the Attorney-General, but is stuck with one...I like that idea...
 
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