Following the discussion at #2037 (comment), I'd suggest that we consider to actively coerce all structured-grid fields to 4D (time, depth, lat, lon), and then allow for any of the dimensions to have length-1. These length-1 dimensions don't have to coordinates.
Then, in the index_search, any length-1 dimension will return a (0, 0) for grid-index and barycentric coordinate. Note that a custom interpolator may then have to be provided
Note that this is effectively already how fields without depth or without time are treated. We would then extend it also to longitude and/or latitude