I recently folded an initial test model with a diagonal shift. At that point, there were still quite a few problems with my test model, including a combination of mathematical complications and folding difficulties. Since then, I have made several changes that help solve those problems without significantly changing the appearance of the final models:
To make the math easier, I created an Excel spreadsheet that automates most of the calculations based on the distances and angles between each fold. This is the first of my models where I have relied on the computer to help figure out the dimensions. I also made a change to the crease pattern that simplified both the folding process and the math.
In my new folding process, the diagonal shift creates a half-twist in the paper, so the paper on the far left above the diagonal shift ends up on the far right on the bottom half of the model. The amount the top and bottom halves are shifted along the diagonal is related to how steep the diagonal is. When the diagonal is close to horizontal, there is very little shift. As the diagonal gets steeper, the amount of shift increases.
I am planning on incorporating this design element into more complex models and hopefully posting some crease patterns soon.