The most likely (but not definitive) cause of smoke in the cockpit would be an electrical issue.
Some questions that may possibly be worth asking to get a fuller picture:
1. Was there any smell at the time? (And did you try sniffing under panel afterwards?) If there was smoke where you saw it, it may not necessarily be the magneto/starter switch, it may have come from somewhere else behind the panel and just appeared there.
2. Were you stationary at the time or did it happen during taxi?
3. Was the door unlatched during taxi? Storm window?
4. When you said it stopped on shutdown, was that shutdown of the engine or master? Did smoke stop with engine stop or master off? Did it stop immediately when doing this?
5. Were there any systems not working normally during the flight? (radio, cockpit lights, popped circuit breakers etc)
6. Did you notice the behaviour of the ammeter during the flight and prior to shutdown? If so anything unusual?
7. Were the air vents open? Was the cockpit fan on? Has arcraft been used for scattering someones ashes recently? (serious question - people do this by opening the storm window and trying to empty it out through that, will result in a blizzard of dust inside the cockpit that will go everywhere and not easily cleaned out. The loved one will need to be hoovered.).
8. weather conditions? (was it humid?)