Colorize material in math-mode subscript and superscripts

You are looking for 'math active' codes:

\documentclass{article}
\usepackage{color}

\newcommand{\amat}{\mathbf{A}}
\newcommand{\bmat}{\mathbf{B}}
\newcommand{\cmat}{\mathbf{C}}

\begingroup
  \catcode`_=\active
  \catcode`^=\active
  \gdef_#1{\sb{\begingroup\color{magenta}#1\endgroup}}
  \gdef^#1{\sp{\begingroup\color{cyan}#1\endgroup}}
\endgroup
\mathcode`\^="8000 %
\mathcode`\_="8000 %
\catcode`\^=12 %
\catcode`\_=12 %

\begin{document}

Some ^_ text.

\(
  \cmat_{i}^{k} = \amat^{i}_{j} \bmat^{j}_{k}
\)

\end{document}

This works as in math mode any character with mathcode "8000 is treated as an active char. The global definition of the behaviour of _ and ^ when active therefore applies even though in text mode they do nothing special.

Here's a LuaLaTeX-based solution. The main input syntax requirements are that (a) the sub- and superscript terms are always encased in curly braces and (b) the arguments of _ and ^ immediately follow these characters, i.e., intervening whitespace is not allowed. It is also assumed that if _ and ^ occur outside of math mode, say, in a URL string, these characters are not immediately followed by a left curly brace.

The solution doesn't modify the catcodes of _ and ^. Instead, it consists of a Lua function that uses Lua's powerful string.gsub function to perform the actual colorizing work and two LaTeX macros, named \sbspcolorsOn and \sbspcolorsOff, that activate and deactivate the operation of this Lua function.

\documentclass{article}
\usepackage{luacode,xcolor}
%% The Lua function 'sbspcolors' does most of the work:
\begin{luacode}
function sbspcolors ( s )
  s = string.gsub ( s , "%_(%b{})" , "_{\\textcolor{magenta}%1}" )
  s = string.gsub ( s , "%^(%b{})" , "^{\\textcolor{cyan}%1}" )
  return ( s )
end
\end{luacode}

%% Two helper macros to activate and deactivate the Lua function:
\newcommand{\sbspcolorsOn}{\directlua{luatexbase.add_to_callback(
   "process_input_buffer" , sbspcolors , "sbspcolors" )}}
\newcommand{\sbspcolorsOff}{\directlua{luatexbase.remove_from_callback(
   "process_input_buffer" , "sbspcolors" )}}

%% A few math-mode macros
\newcommand{\amat}{\mathbf{A}}
\newcommand{\bmat}{\mathbf{B}}
\newcommand{\cmat}{\mathbf{C}}

\begin{document}
$\cmat_{i_{u}}^{k^{\ell}} = \sum_{j=1}^{k} \amat^{i}_{j^{M}} \bmat^{j_{N}}_{k}$ \quad
\sbspcolorsOn  % activate the Lua function
$\cmat_{i_{u}}^{k^{\ell}} = \sum_{j=1}^{k} \amat^{i}_{j^{M}} \bmat^{j_{N}}_{k}$ \quad 
\sbspcolorsOff % deactivate the Lua function
$\cmat_{i_{u}}^{k^{\ell}} = \sum_{j=1}^{k} \amat^{i}_{j^{M}} \bmat^{j_{N}}_{k}$
\end{document}