To find iterative methods is an amusing pastime. While shopping with wife and daughter I found a nice decomposition of rhombs, squares and octagons. Their sides are then reduced by the factor 2+sqrt(2) = 3.14, which is not related to the corresponding factor 1+sqrt(2) of the Ammann-Beenker tiling. Thus, this is a new tiling and not a mere redecoration of the squares and rhombs of the Ammann-Beenker tiling. For simplicity I use only one single color for each tile.
The new tiling has rings of octagons connected by corner points:
Because of self-similarity and arising from the iterative method most rings of eight octagons are again connected to form hyper-rings, and so on.