Shell – Renaming the current directory from a shell – possible

Best would be to use zsh's zmv:

autoload zmv # best in ~/.zshrc
zmv -n '(*)/index' '$1/C2_$1'

(remove -n when happy).

For a portable (POSIX sh) solution:

for dir in */index;  do
  mv -i -- "$dir" "${dir%/*}/C2_${dir%/*}"
done

(using -i as a poor man's ersatz to the sanity checks zmv does).

If you wanted to use find portably (POSIXly), you'd need to forget about -mindepth/-maxdepth, which you can replace with combinations of -path and -prune:

LC_ALL=C find . -path './*/*' -prune -name index -exec sh -c '
  for dir do
    top=${dir#./}
    top=${top%/*}
    mv -i -- "$dir" "$top/C2_$top"
  done' sh {} +

One difference with the other two approaches is that it will not follow symlinks and that it will also look for index in hidden directories.

Those make use of the standard ${var#pattern}, ${var%pattern} parameter expansion operators described in countless Q&As here or at the POSIX shell specification.

I would go about this in a different way - specifically, using a depth-first search in place of the shell globstar **

For example, using GNU find, given:

$ tree
.
└── dir
    ├── foo
    │   └── baz
    │       └── MainFoo.c
    └── Foo
        ├── baz
        └── MainFoo.c

5 directories, 2 files

then

find . -depth -iname '*foo*' -execdir rename -- 's/Foo/Bar/;s/foo/bar/' {} +

results in

$ tree
.
└── dir
    ├── bar
    │   └── baz
    │       └── MainBar.c
    └── Bar
        ├── baz
        └── MainBar.c

5 directories, 2 files