What does it mean?
This diagnostic trouble code (DTC) is a generic powertrain code, meaning it applies to vehicles equipped with OBD-II. Although generic, specific repair steps may vary by make/model.
This fault code essentially means the PCM/ECM (Powertrain/Engine Control Module) has detected a fault in the internal control module’s ROM (Read-Only Memory) within the PCM. The PCM is essentially the vehicle’s “electronic brain” controlling functions such as fuel injection, ignition, etc. When self-checks fail in the ROM, this DTC is set.
This code is a generic powertrain code. It is considered generic because it applies to all makes and models of vehicles (1996-newer), although specific repair steps may be slightly different by model. A quick online search shows this DTC is more commonly found in Ford and Nissan vehicles.
Other internal control module error codes include:
P0601 Internal Control Module Memory Checksum Error
P0602 Internal Control Module Programming Error
P0603 Internal Control Module Keep Alive Memory (KAM) Error
P0604 Internal Control Module Random Access Memory (RAM) Error
Photo of a PCM with the cover removed:

Powertrain Control Module PCM
Symptoms
Symptoms of a P0605 DTC will include MIL (Malfunction Indicator Lamp) illumination, though there may be other symptoms including but not limited to various warning lights on the dashboard, stalling, and no-start conditions.
Potential Causes
There is an internal fault in the PCM/ECM
Using an aftermarket performance programmer may trigger this code
Possible Solutions
In some cases, “flashing” the PCM with updated software may fix this DTC. You will need access to make and model specific information such as Technical Service Bulletins (TSBs).
If there are no PCM flash updates, the next step would be to check certain wiring. Inspect and verify the voltage and grounding of the PCM and related circuits. If any issues are found, repair and retest.
If wiring is OK, the next step is to replace the PCM, which is the most likely repair for this code. This is typically not a DIY task, although in some cases it may be. We highly recommend having the work performed by a qualified repairer/technician who can reprogram the new PCM. Installing the new PCM may involve using special tools to program the vehicle’s VIN (Vehicle Identification Number) and/or anti-theft information (PATS, etc.).
Instead of replacing the PCM, some specialized shops may actually be able to repair the PCM. This may involve removing the PCM, shipping it to them for repair, and reinstalling it. This is not always an option for daily drivers.
NOTE: This repair may be covered under the emissions warranty, so be sure to check with your dealer as it may be covered beyond the bumper-to-bumper or powertrain warranty period.
Other PCM DTC codes: P0600, P0601, P0602, P0603, P0604, P0606, P0607, P0608, P0609, P0610.