Just got done taking quantum mechanics this semester as my last technical elective towards a mechanical engineering degree. The primary driver for theories describing the universe in more than four dimensions is that gravity is not understood very well, even with the special theory of relativity. Quantum physics mathematically describes electricty, magnetism, and the strong and weak nuclear forces with incredible accuracy. Many of the values predicted for physical constants of the universe match those obtained in experiments far more accurately than classical physics ever did. (Some to 20 decimal places or more). The equations of quantum mechanics can be simplified or reduced to the equations of classical physics when classical assumptions are made.
But when quantum mechanics is applied to gravity in a 4-dimensional universe, it doesn't work. It predicts gravity (and the gravitational constant) should be a lot stronger than what we observe. (By many orders of magnitude). One mathematical explanation for this is extra dimensions. As simply as I can state it, the extra gravitational force "bleeds off" into these extra dimensions. Also, since scientists have yet to observe any force carrier for gravity, gravity wave, or any subatomic particle from which the property of mass arises, it would also be possible that this originates in an extra dimension, and gravity and space-time curvature are the only effects that we are able to observe (or as yet know to look for) in our 4-dimensional universe.
The effort to understand gravity has led to the string, super-string, and brane theories that are around today that all have more than 4-dimensions. Theories like alternative realities have been made popular by sci-fi but are mostly fringe theories and are more based on new age metaphysics than mathematics or the scientific method. Some people have taken things like the uncertainty principle, superposition, entanglement, and probability, and ran with it in all sorts of philosophical directions without really understanding what the underlying mathematics means.
Like a previous poster mentioned, anything that is real (in the context of what I think the OP means) in a dimension in our universe would be real in all dimensions, but not necessarily observable, though its effects might be if one knows what to look for. Brane theory skews that idea somewhat but without getting technical the result from our point of view would be the same.
Again as mentioned "real" is a relative term. For example, AC electricity has both real values and imaginary values. The imaginary part can't be measured by an electric meter. But the imaginary part of the electricty costs real money for the electric company to produce. That part of your electric bill is estimated by a multiplying factor. Something else intriguing are "virtual particles" which are basically not real but can become real, and are used in the explanation of hawking radiation emitted by black holes.