I guess it depends on what you mean by power, but by the standards of what computation theory generally considers power, this is a somewhat bizarre question, for how commonly it seems to arise.
In a Turing machine model, all computing is reducible to something 1-dimensional with no loss of generality. (2 if you are counting time, and reducing the Turing machine to a tiling of something.) So dimension disappears from consideration.
And relative to "Big O's" it also lacks some sense. Computer connections are not propagated in any particularly direct spatial manner, two signals can take equally long to reach the same place even if one follows natural space and the other uses a slightly better conduit. From a direct efficiency point of view, therefore, it is graph-theoretic dimension that matters, and any graph-theoretic dimension can be modeled in 3D space. Spatial dimensions can decrease the multiplier in a speed calculation, but not the order of growth.
From the points of view where dimensionality does bear on the question, is not really an aspect of computing, but of mechanical engineering as applied to it.
From another direction, 4 spatial dimensions would make for a lovely universe, but it is not ours. Folks become strongly attached to the notion of 4D space before they realize that basic Newtonian physics indicates we cannot have four macroscopic dimensions. If we did, we would encounter some basic field effects that decrease as the cube of the distance from their origin. These should be numerous. But we never see them.
And we would have to make really serious reasons for why radiant energy itself would not be an inverse-cube field effect. Energy should not obey an inverse-square law in 4D space. Given that, if we have any additional dimensions, they are all either microscopic (curved far too tightly to have any use for computer engineering) or temporal.