Is a black hole a three dimensional or four-dimensional hole?
It would help to clarify your question on what exactly you mean by a 4th dimension. Space-time for example is 4 dimensional everywhere, but it's 3 spatial dimensions and 1 time dimension.
Quantum theory allows for extra dimensions, but they're very small, smaller than an atom and impossible to measure by normal means. CERN has looked for those folded dimensions but has found no evidence of them, and string theory defines 10 or 11 dimensions, most of them we can't see or measure, if they exist at all.
If you mean 4 spatial dimensions, then the answer is no, with a capital N. There's no evidence that an extra spatial dimension would form in a black hole. There's nothing in relativity that suggests that might happen. An extra time dimension . . . probably not either, but it's fun to think about what 2 time dimensions would look like.
What happens inside a black hole is that one of the dimensions becomes like time. It only moves in 1 direction, but that doesn't make it a 4th dimension, it just changes the nature of 1 of the dimensions.
If you actually had a 4th dimension (and the book flatland discusses this in pretty good detail), but if you actually had that, very strange things can happen, for example, you can be flipped over, like flipping a card over, which can represent 2 dimensions, from face down to face up. Your left side would become your right side. Your heart would be on the other side of your chest and all the left handed molecules in your body would become right handed molecules, which would make your ability to absorb and use nutrients that you digest problematic. It might even make things taste and smell different.
You can't tie your shoes in the 4th dimension, but even if you could, your shoes could slide off your feet, no matter how well you tied them. Food could fall out of your stomach, not through one of the two normal exits, but it could just slip out, like a ball rolling off a table. Anything not attached could slip apart. Sex would be much more difficult. Bolts and nuts could just move sideways and detach, unless there was some electrostatic charge keeping them in place, in fact, the pressure that forms natural between a screw and a nut might cause all nuts and bolts to fly apart at considerable velocity once you entered 4th dimensional space.
Two people could also stand in exactly the same place, just in a different 4th dimensional space. That brings a whole new aspect to the how many people can you fit in this Volkswagen game, as you could fit theoretically any number of people in the same Volkswagen.
A 4th spatial dimension is probably impossible, because it breaks too many rules and laws as we know them, and it's never been observed.
Black holes, by comparison, can do strange things like slow down time, and they aren't fully understood, but a 4th spatial dimension probably isn't possible, unless you mean at quantum distances, in which case . . . maybe.
Feel free to read this for a different approach to a somewhat similar question.
Given the current state of knowledge, the answer is yes: a black hole exists in three space dimensions plus one time dimension.
The spacetime around a black hole is often depicted using the rubber-sheet model, e.g. Flamm's paraboloid for the Schwarzschild metric. This makes it look like spacetime is curving through a higher-dimensional space (i.e. the 3D space containing the sheet) but you have to bear in mind that this is just a visualisation. In order to make it comprehensible it needs to introduce an imaginary extra dimension outside of spacetime because humans are in general not equipped to visualise the geometry involved thanks to our evolution and experience in a region of the universe where space is near-Euclidean and time can be treated in everyday situations as being independent of space. There is at present no evidence that our 3+1-dimensional universe is embedded in a higher-dimensional space.
There is of course the possibility that whatever's actually going on (that general relativity and quantum field theory are merely very good approximations of) does involve extra dimensions. Extra dimensions have been invoked by several theoretical attempts to find a unified "theory of everything" but so far getting evidence for these theories being correct has proven extremely difficult. It doesn't help that the Large Hadron Collider served up the blandest Higgs boson on the menu, then proceeded to (so far) not find any evidence of things like supersymmetric particles that may have proven that these kind of theories were on the right track.