View source for 2^4

Many of the techniques used on the 3x3x3x3 also apply to the 2x2x2x2.
==Cell By Cell Method==
This is the 4-dimensional equivalent to the Layer By Layer method on the 2x2x2.
===Solve a cell===
This can be done either by just orienting the cell first, then putting that cell at I and using -O moves to permute the pieces of it around, or by using direct blockbuilding to solve the pieces.
===Orient Last Cell===
Similar to OLL, in this step the solver orients all 8 corners of the U cell so that the U colour is on the U cell. This can be done with RKT setup moves to create familiar 2x2x2 OCLL cases, then 4d rotating to use RKT on a different cell to orient those corners. Another strategy is to pair up 2x2x1(x1) blocks or oriented corners, and permute those using -U variants of OCLL algorithms.
===Permute Last Cell===
For the final step, permute the last cell just like a regular 2x2x2 Rubik's Cube. You will have to use RKT to do this. There is a 50% chance of RKT parity occurring, which you will need an algorithm to fix.

==Ortega Method==
This is the 4-dimensional equivalent to the Ortega method on the 2x2x2.
===Orient a cell===
Orient a cell.
===Orient Last Cell===
Similar to OLL, in this step the solver orients all 8 corners of the U cell so that the U colour is on the U cell. But because the First Cell was not permuted, you don't have to use RKT in order to orient it (though it can be certainly helpful), which saves a few moves.
===Permute Both Cells===
Now just permute both of the cells like normal 2x2x2x cubes. For the first cell, you don't have to use and RKT because you can just put the first cell at I and use -O moves to turn its sides. For the Last Cell, you will have to use RKT.

==Luna's Physical 2x2x2x2 Method==
This method works really well on Melinda's Physical 2x2x2x2 puzzle [https://superliminal.com/cube/2x2x2x2/] , but it can be used on the virtual 2x2x2x2 as well. 
===Domino Reduction===
Orient 2 opposite cells of the puzzle, but they can be oriented with either colour.
===Permute First Cell===
Treating the whole puzzle like a 2x2x4 domino puzzle (except you can rotate the U/D halves in any direction), solve a cell. This will automatically complete separation of the U/D colours.
===Permute Last Cell===
For the final step, permute the last cell just like a regular 2x2x2 Rubik's Cube. You will have to use RKT to do this. There is a 50% chance of RKT parity occurring, which you will need an algorithm to fix.