Hey guys, I wanted to share a new concept for twisty puzzles that I came up with over the past month. It may have already been discussed but I'm pretty sure it's new.

The concept I'm going to share is one I call "multi-bandaged puzzles." Here is the idea: imagine a twisty puzzle that has a bandaged subset (for example the Master Skewb, it's bandaged subset is the Offset Skewb):

Now imagine that the bandaged subset is taken, and the unbandaged puzzle is built around it, like a multi-puzzle:

The way such a theoretical puzzle would function is like this: the inner and outer puzzle are connected in such a way so that if the outer puzzle can turn one way and the inner puzzle also can turn in the same way, both puzzles will turn:

However, if the outer puzzle can turn one way and the inner puzzle cannot turn in the same way, only the outer puzzle turns, like this:

It seems as though such a puzzle would be much more difficult to solve than it sounds, because pieces which may seem as though should behave in the same way really don't since the outer puzzle has many more ways to turn than the inner puzzle.
I already have explored other puzzles where this idea could be implemented, for example the deepcut FTI (as its subset is a Skewb) or even a 12 segmented 3-layer cheese (its subset is a square-1!)
Keep in mind that I see this puzzle idea as a solving concept, not as a design challenge (I don't even think it's possible to build). I'm also wondering if it's possible to simulate such a puzzle in a program.
I'm interested in hearing your thoughts on this idea. Thanks!

What you've come up with is very closely related to the idea of a "Complex Puzzle" and your example is really a subset (of a subset) of the pieces from the Complex Vertex-Turning-Cube.

Take the Offset Skewb for a moment. I think you'll agree that the puzzle as a whole has 4 total turnable portions (grips) and a Skewb has 8. Thinking mechanically, the 4 vertices that can't turn, can't do so because pieces physically block them. You can imagine abstractly turning those grips even though no external pieces would move. They wouldn't move because none of the external pieces on the puzzle make use of any of those 4 grips. By layering a Master Skewb in parallel with the Offset Skewb you've added pieces that do move with the other 4 grips so this abstract turning idea now has a visible effect on the puzzle.

The reason I say what you've made is a "subset (of a subset) of the pieces from the Complex Vertex-Turning-Cube" is that you've left out half of the new Offset Skewb pieces.

Instead you'd want two Offset Skewbs, one using one set of the 4 vertices and the other using the other 4. The Master Skewb would use all 8. This puzzle would consist of a subset of pieces from the Complex Vertex-Turning-Cube. By leaving out one of the Offset Skewb you've left out half of one of the new pieces types (you left out a whole orbit).

Take a look at Gelatinbrain's 3.3.20 puzzle which is a Circle Skewb for an example of a puzzle that has both sets of Offset Skewb corners.

The more layers of odd turning you add the closer and closer you get to the Complex Vertex-Turning-Cube which would be a very hard puzzle to solve.