Can ',,' be aliased to '..'?

ZSH:

If you'r using zsh alias -g ,,=".." is what you need, but this will cause ",," to be expanded everywhere, not only when used with cd.

From man zshbuiltins:

If the -g flag is present, define a global alias; global aliases are expanded even if they do not occur in command position.

BASH:

If restricted to bash (as the question is tagged with bash), read the following:

This is a pure-alias solution, as requested, however this will do more than required possibly frustrating the user (see the warning at the end of my post).

Quoting man bash:

If the last character of the alias value is a blank, then the next command word following the alias is also checked for alias expansion.

Therefore it's enough to alias cd with an extra space (to allow expanding of its next argument) and then alias ,, to ... See that

alias cd='cd '
alias ,,='..'

gives exactly what you need.

Note that this is correct not only for bash (and its alias implementation), but all POSIX-compilant shells. Quoting an example from man 1p alias (the manual does not describe this feature explicitly, only through an example):

4. Set up nohup so that it can deal with an argument that is itself an alias name:

          alias nohup="nohup "
   

Warning: As @PeterCordes writes in his comment, it will automatically cause other aliases to expand when written after cd. It may require you to write cd \grep if you want to change directory to one named grep but your grep is an alias for grep --color=auto. Without the backslash, cd will report "too many arguments" error (you can't cd to two directories at once)!.

Aliases aren't meant to do this, but you can create a function called cd that is a wrapper for the real cd. This one works for me! Just put it into your .bash_profile or your profile file of choice.

cd () { if [ "$1" = ",," ]; then builtin cd ..; else builtin cd "$@"; fi; }

Or, with comments and pretty formatting:

cd ()
{
  if [ "$1" = ",," ]; then  # if first argument is ",,"...
    builtin cd ..           # call the builtin cd with ".." instead...
  else
    builtin cd "$@"         # else call builtin cd with original arguments 
  fi
}

EDIT

Thanks @derobert for pointing out that if then else is better here than && ||. Also, I just realized (*facepalm*) that this implementation will only work if any non-path arguments (i.e. -L -P) are not the first argument. So be warned, if you have a bad habit of typing cd -L ,, this isn't going to help you. And that is non-trivial to handle correctly, I think.

Aliases must be the first word of a command. Also, the alias must be substituted for a word therefore no spaces).

Bash Reference Manual: Aliases

Aliases allow a string to be substituted for a word when it is used as the first word of a simple command.

You could alias both .. and ,, to be cd ...

$ alias ..="cd .."
$ alias ,,="cd .."
$ cd /tmp && pwd
/tmp
$ ,, && pwd
/

Another way to make .. change the directory is to enable Bash's autocd feature:

shopt -s autocd 

That'll automatically change the directory when you enter the name of one:

~$ /tmp
cd /tmp
/tmp$ 

You still have to make an alias for ,, but that can be done with just alias ,,=..