Let's imagine we have an arbitrary 1-qubit quantum system $\alpha \vert 0 \rangle + \beta \vert 1 \rangle$ Making a measurement in the +/- basis is equivalent to performing a Hadamard gate and then making a measurement in the standard computational basis.
Can we extend this notion generally for measurement of an arbitrary 1-qubit state in any arbitrary orientation? So, if we want to make a measurement of an arbitrary 1-qubit quantum state in some arbitrary orientation in the 2-dimensional Bloch sphere, can we perform some unitary transformation $U$ (supposing such $U$ can be composed theoretically) that transforms the basis state into orthonormal states of that orientation and then perform a measurement in the computational basis?
Let's say we have unitary transformation $U$ that maps $\vert 0 \rangle$ to $\vert a \rangle$ and $\vert 1 \rangle $ to $\vert b \rangle$ with $\langle a \vert b \rangle = 0$ (where $\vert a \rangle$ and $\vert b \rangle$ have certain orientation with respect to the standard basis states) the same way $H$ transforms $\vert 0 \rangle$ to $\vert + \rangle$ and $\vert 1 \rangle $ to $\vert - \rangle$ state. One difference could be that $H$ is Hermitian too while $U$ may not be. Another sub question is whether $U$ needs to be Hermitian also to make this generalization.