If I understand correctly, we see the electron appear as a point on the detector in the double slit experiment because the position basis vectors are orthogonal. This ensures that we will only see one of these possible outcomes due to decoherence. However, isn't it possible to decompose the wavefunction of position differently? Why can't we make the basis vectors some combination of locations so that we observe some combination of points appearing on the detector?
When a measurement is made, the wavefunction collapses to an eigenstate of the operator corresponding to the variable being measured. While one certainly can express a wavefunction in a basis of states equal to some linear combination of position eigenstates, the elements of this basis will not be eigenstates of the position operator, so they will not be obtained by measuring the position.