Pulse oximeters are useful for measuring oxygen saturation in the blood. They have no capability to measure pH or carbon dioxide. In fact, that is a potential problem with them. People can have fairly normal oxygen saturations because of compensatory hyperventilation. So what looks good could in fact can be bad e.g., oxygen saturation is maintained near-normal at the expensive of hard respiratory work cause by an underlying pulmonary problem. You would know this by measuring the respiratory rate or, if you could, by measuring pH and/or carbon dioxide levels. There are other caveats as well e.g.; the interference caused by carbon monoxide.
Assuming that the person has good skin perfusion, a pulse oximeter works the same at altitude as it does at sea level. As ambient atmospheric pressure decreases, the amount of oxygen in air decreases correspondingly. Respirations increase to help compensate for this. In general, however, resting saturations will continue to decrease as one ascents to a higher altitude.
The device itself is not a good tool for diagnosing HAPE/high altitude pulmonary edema. It could give you the impression that things are better than they really are because of what I mentioned above. A history of a person’s activity level (decrease in exercise tolerance), vital sign assessment including mental state and lung exam with a stethoscope are in the end cheaper and more accurate.