I wonder if it could be as simple as a failing ignition switch. If the engine electronics are powered up while the engine is not running, the CEL is on - at least some of the time. How well does the behavior match what you would see if you turn the key to the "run" position, but don't start the engine. We've seen ignition failures before, though those were all along the lines of accessories/windows/etc. don't always work when the car is running.

Have you tried pulling fuses to see if/which one extinguishes the CEL? That and some time staring at a wiring diagram could start to narrow down what is causing the CEL to illuminate.

My rambling $.02,

- JimY