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Mt. Everest is higher. Because of the Earth's rotation, every point on earth traces 360 degrees every day. However the radius of these arcs depends on elevation. Ergo the top of Mt. Everest travels farther every day due to rotation that any other place on earth. That is the same as saying it is moving faster.

EDIT: I guess compared to a rotating reference frame this isn't true? Clearly this isn't my area of expertise.

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Not that I disagree with your point in any way, but I thought you might like to know - the top of Everest is not the furthest point from the centre of the earth. It is the highest point above sea level, but given the slightly ellipsoid nature of the planet, the spot with the largest radius value is actually in South America - http://en.m.wikipedia.org/wiki/Chimborazo

I think a way to rephrase the question is to ask, is the light coming from the top of mt. everest red shifted or blue shifted or neither? (I'm not certain of the answer, either. Physics was a long time ago.)

It depends on the location of the observer. (coming towards you = blue, moving away from you = red)

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