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Is "altitude" an absolute measure of air "thinness"?

I am wondering if altitude is the only thing you need to know in order to evaluate the "thinness of air" at certain point on Earth. For instance, will climbing a 6000 m peak in the Andes or a 6000 m peak in the Himalayas make any difference in how you need to acclimatize to the elevation? (I'm not referring to how tough the climb is.)