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In 3 spatial dimensions, $$[E] = [ML^2 T^-2]$$

Would it change in higher dimensions? If yes, then what would be the dimensions for 4 spatial dimensions?

Suppose for a moment that we're specifically interested in the kinetic energy of a single, non-relativistic particle, so that $E=frac12mvecv^2$. I include the vector notation for the velocity here because it's useful to keep in mind that $vecv^2=vecvcdotvecv$.

If the particle is moving in two dimensions, then $vecv=v_xhatx+v_yhaty$ and $vecvcdotvecv=v_x^2+v_y^2$, which has units of velocity squared. In three dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz$ and $vecvcdotvecv = v_x^2+v_y^2+v_z^2$, which still has the units of velocity squared. In four dimensions, $vecv = v_xhatx+v_yhaty+v_zhatz+v_whatw$, and $vecvcdotvecv=v_x^2+v_y^2+v_z^2+v_w^2$, which, again, still has the units of squared velocity. This is because adding two quantities of the same units together does not change their units. Indeed, this holds for any number of dimensions, and suggests that the units of energy should be unchanged.

We can also see this more generally by the formal definition of work, which is the definition of change in energy:

$$W=int_C vecFcdot dvecs$$

for an object being acted on by a force $vecF$ moving along a path $C$ with arclength parameter $dvecs$. This is the formal, most general definition, and holds no matter how many dimensions of space you have. Once again, you can see that this definition involves a dot product. The dot product takes two vectors in any number of dimensions and outputs a one-dimensional quantity (i.e. a number). The work only cares about one component of the force, namely, the one pointing along the one-dimensional path that the particle is taking. No matter how many spatial dimensions this one-dimensional path is embedded in, the work only cares about things happening along one of those dimensions. As such, the units of work should always be the same, and, since work has the same units as energy (otherwise we wouldn't be able to add them together), the units of energy should also always be the same.