Global phases are physically irrelevant
If a qubit is multiplied by an overall global phase: $$e^{i\theta} \left(\alpha \ket 0 + \beta \ket 1\right),$$ for some angle $\theta$, it has no effect on the outcome of the measurement. The probability of getting a certain state after measurement would be the same as it were without the global phase. This is true no matter what measurement basis we use. 1