I played with C^4 and I conjecture that the tools from C^3 are
sufficient to solve it. I have not completely finished it but I
think it is just a matter of time.

One transformation I use repeatedly, in generic terms, is
S = top 180, 2nd slice up or down, top 180, 2nd slice down or up. This
is the verticle slice which is facing you. This is just the move in
C^3 to move 3 edges in a plane whereas in C^4 the outcome is far more
complicated.

The steps for C^4 are then
1) Align the centers with a map. Some use of S is necessary.
2) Do all corners as in C^3.
3) Use S repeatedly to position the edges and this is very laborious.
4) Use the C^3 edge flip (Rubik's transformation) to finish it off.

After several hours of C^4 I find C^3 looks like a toy.