Outside though, you'd see everything start to blue-shift. Things below you would blue shift back to normal, and the universe above you would blue-shift and speed up until you'd see the heat death of the universe. Anything falling in after you would red-shift again as it approached to match your "normal" rate of time. Critically this would include any light or other particles, so it might be very hard to survive.
No matter how fast you go or how weird the space time you are in, your local clock should still tick steadily to you, and you wouldn't notice anything weird.