Will I hyperventilate if I breath twice as fast at an altitude with half as much oxygen as I am used to?
By definition, hyperventilation is a state of increased breathing where the exhaled $CO_2$ is greater than what is produced by the body.

Except in artificial condition or in disease process, breathing faster than the autonomously-dictated rate is going to cause hyperventilation. You have a fine-tuned control of respiration that will work out the correct breating rate for you.

Oxygen is actually irrelevant for this phenomenon, first because of the definition of hyperventilation, but also because oxygen is mostly transported through haemoglobin (98+%). This means there is a fixed quantity of oxygen that can be transported by unit volume of blood (and by breath, for a given cardiac output), and the maximum is already reached at normal breathing rate. Breathing faster does not allow for better oxygenation of the blood. (even at an higher altitude)

On the other hand, $CO_2$ is transported by the plasma, dissolved (7%), but also in the form of bicarbonate (93%) through the following equilibrium :

$$ H_2O + CO_2 \rightleftharpoons \underbrace{H_2CO_3}_\text{Carbonic acid}  \rightleftharpoons \underbrace{HCO_3^-}_\text{Bicarbonate} + H^+  $$

In the alveoli, a quantity of $CO_2$  will diffuse to the alveolar air, and be breathed out, the target being eliminating the quantity produced by the body. However, in opposition to oxygen transport, it is possible to overshoot by breathing faster, because removing $CO_2$ drives the equilibrium in direction of $H_2O + CO_2 \leftarrow  HCO_3^- + H^+ $ that is, eliminating additional $CO_2$ through the removal of bicarbonate from the blood.

This is hyperventilation, and leads to hypocapnia, causing that particular sensation of dizzyness. It is also leads to respiratory alkalosis, because, as seen in the reaction equation, this incurs the loss of protons ($H^+$).