The computer and the alternator warning light have no logical connection.
Dumping the codes will not be useful in finding the alternator problem.
The most likely problem is a fault in the internal voltage regulator. The alternator
warning light serves to feed a signal to the alternator to "turn on" and start
making electricty. Remove the bulb and resistor and a perfectly good alternator
will not put out any power.
As shown in the diagram, alternator warning light gets its power feed from the
ignition switch and is energized anytime the ignition is in the Run position. Under
normal circumstances, when the ignition is in the Run position and the engine is
running at sufficent speed, the alternator warning light goes out because the
voltage regulator turns off the ground that is needed to make the bulb glow.
When the voltage drops below some set point, the voltage regulator turns on the
ground on alternator warning light circuit and the bulb lights up.
See the following website for some help from Tmoss (diagram designer) &
Stang&2Birds (website host) for help on 88-95 wiring
http://www.veryuseful.com/mustang/tech/engine/
http://www.veryuseful.com/mustang/tech/engine/images/IgnitionSwitchWiring.gif
http://www.veryuseful.com/mustang/tech/engine/images/fuel-alt-links-ign-ac.gif
Diagram courtesy of Tmoss & Stang&2birds