Jared is right, I mean cube which has all movements of normal Mixup (normal shape) + (always or sometimes?) allows movements if only one layer is turned by 45 degrees.
Can this cube be implemented?
Yes and no...
If I add the word "always" to your question the answer is no. The reason being that after a single 45 degree turn of a face the puzzle is no longer in a doctrinaire position. If you allow a turn at this point the puzzle jumbles. To "always" allow this turn you must completely unbandage the puzzle such that its doctrinaire again and if you try this process on this puzzle you end up cutting the puzzle up into dust.
If you want it to allow this type of movement "sometimes" you can stop the unbandaging process after a number of steps and be left with a puzzle which jumbles but after some number of turns you will find that some moves become blocked. Even if you stop this process very early I think in practice you will find that you've created so many new pieces that it will be hard to make a stable puzzle.