Technically, you can have a supercharger and turbocharger installed at once. This is called twin-charging, and it is mainly used to reduce or completely eliminate turbo lag. The supercharger is used at lower engine speeds when the turbo is spooling up, then the turbo is used to provide a majority of the boost.

There were (and still are) a few cars that used twin-charging, and I believe Mazda's new "sparkless" engine concept is using twin-charging to increase cylinder compression for auto-ignition like a diesel. 

In essence, it will be VERY COMPLICATED to twin-charge your ZL1 Camaro, and it will cost an insane amount for the installation, dyno testing, re-programming, and miscellaneous custom work done to the engine to support both a supercharger and turbocharger. The engines that come with twin-charging from the factory are already way too complicated to work with.

Do you have $80k - $100k for modification? Twin-charging is extremely expensive to do correctly and puts a ridiculous amount of strain on the motor. You'll need to pull the brand new motor, fully forge and build it out to handle the pressure, pull and build the transmission to hold it, the rear end.....you are talking BIG money. 

Twin turbo would make more sense because you could program one smaller turbo to be in full boost to about 4500 rpm and then have a bigger turbo pick up and pull it to redline. How much power are you trying to make? You'll need a lot of suspension changes as well to handle launching the car....just a TON of work any way you cut it.

Just to be honest, I've built a few twin-charged custom cars. NONE were done for under $250k and all were done for major car shows. (This is for a fully custom, ground up twin charged vehicle.) Unless you're trying to accomplish something specific, I don't  see why you would want to take a brand new car and chop it up. Being a GM I wouldn't do ANYTHING to void the warranty.

If you want a big power car, I'd get an older car for a few grand and just build it how you want it. You're looking at $80k for the new ZL1 and another $80k - $100k to twin charge it. Not worth it.

That is not a good idea.  The Camaro ZL1 is already complicated enough of a vehicle as is and being a GM product with compromised reliability, why push your luck further by adding a turbo charger?  I would leave as is.  Keep the vehicle under warranty and get rid of before the warranty expires.