We know that Bell's theorem implies that there cannot be any local hidden variable behind quantum phenomena, right?
If we try to explain it using non-local hidden variables, can it be explained as local with respect to another spatial and time dimension?
what do you mean by "..another spatial and time dimension"?? are you implying to use an extra set of dimensions apart from the 3+1 dimensions?? or are you suggesting in another frame??
However any non-local phenomena (any phenomena explained by non-locality is non-local) will remain non-local in any frame. Quantum mechanics is essentially non-local in nature. A very simple example that I like to give is the very act of measurement and wave function collapse. It all happens "instantaneously" in all reference frames. Any underlying theory (if it exists) which is more fundamental than QM should be non-local.
"Non-local", in the context of Bell's theorem, is not a reference to any specific dimensions or geometry. It is simply a recognition of the fact that the correlation of probabilities between two events in certain experiments is not consistent with the idea that the two outcomes are independent. That is, if I label the two events A and B, the outcome of A cannot depend only on information that excludes knowledge of the outcome of B; and that similarly, the outcome of B cannot depend only on information that excludes knowledge of the outcome of A. Since it is possible to set up A and B such that they are "spacelike separated" (i.e., no classical signal can travel from A to B or from B to A) some kind of a "nonlocal influence" (that is, anything other than a conventional signal propagating at or below the speed of light) is assumed to connect the two. But we also know that the same quantum mechanics that describes these Bell-type experiments also forbids classical violations of causality. Beyond that, you are free to speculate as to what that influence might be, but it's probably a good idea to first establish a good understanding of what they cannot be, otherwise you're likely to travel down roads that have been explored by others and which take you to dead ends.
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