Back in 2007 I created my own method to solve the 4 dimensional 3x3x3x3 cube. It was a dimension-reduction method with lot of 3-cycles for easier tracking of the pieces. Took me 9 hours for a single solve - awfully slow, but it did the job, and I was able to land on the 58th place in the
Hall of Fame.
I also created my own method for the physical 4 dimensional 2x2x2x2. I call it the "Dimension-reductional Ortega" method. Though I am fairly certain that many people in the hyperspeedcube community developed the same or very similar method.
Here are the steps:
1. Force the purple and pink stickers into an "internally oriented" state (i.e., non-outer corner segment).
2. Solve type-1 parity if necessary through a gyro setup.
3. Force the purple and pink stickers into an "externally oriented" state.
4. Separate the purple and pink faces. The hypercube is now reduced to two 3D cubes (apart from later parities).
5. Solve the purple 2x2x2.
6. Solve type-2 parity if necessary using a pink corner piece.
7. Solve the pink 2x2x2.
8. Solve type-3 parity if necessary with a simple alg.
Type-1 parity is when one of the purple/pink stickers misoriented relative to the others.
Type-2 parity is when the reduced 2x2x2s are "unsolvable" in a regular way; one corner from each cube must be rotated simultaneously.
Type-3 parity is when one 2x2 section is rotated by 180 degrees (needs a semi-complicated "U2" AUF).
My very first timed solve with a prototype variant of this method (with slight differences) was 7 minutes: