There's no coaching change, and there might not be. Even if there is, I'd probably push toward a Coryell type system. There would be continuity; it features a downfield game that benefits Cam Newton, Steve Smith, and Greg Olsen. Regardless of what happens with the team, these three are big parts of that.
But, there's no telling what might happen. A new coach might come in, and that coach might have his own offense (or, as a defensive guy, has an OC he prefers, which may mean the new head coach prefers that OC over caring about the offense at all).
So, let's go through the other offenses and find where the fits are.
West Coast Offense
The WCO has some positives - Smith's run after the catch ability, the added emphasis on getting the ball to the backs. It fits into a zone blocking scheme that won't be far off where Carolina falls right now. Mike Shanahan has shown it's a malleable enough offense to throw in the pistol and zone read. There's a lot of moving pocket stuff that would fit.
It's only more recently that shotgun ideals have worked with the WCO. Carolina ran more shotgun than anyone but Detroit (another Coryell, under Scott Linehan). So there would be an adjustment. Olsen, always under a Coryell scheme, would get his touches but probably wouldn't get the downfield stuff he's enjoyed here.
This scheme, most reminiscent of the NY Giants/Patriots lineage, is the most dependent on who's running it. A Jeff Davidson scheme here, it came to the Patriots as Charlie Weis came over with Bill Belichick. The Patriots have used it toward spread ideals lately, featuring the slot receiver and tight end. Steelers disciples generally run this as well.
This isn't really an "offense" that was developed in the same form as the others, in a way. It's just old school football that eventually came together in one playbook, untouched by the innovators of the 70s and 80s. In its pure form, it's a power running, play action offense. This would inevitably limit Newton's passes and touches, and work harder on the run game. Though Dan Henning ran Coryell, his ideals were more of the E-P variety - "feed the stud", pound the ball, take deep shots, play the odds.
Another situation where shotgun isn't an ideal format. More or less wastes Smith, and Olsen isn't blocker enough. The backs would get plenty of touches.
Run and Shoot/Spread/No-Huddle
The spread offenses of the last 15-20 years in college are adaptations of the original run and shoot offense. A few teams in the league run this - though their adaptations can be made from any offense, notably Belichick's spread ideals above. This offense was huge in the AFC in the early 90s, with various no huddle, run and shoot type offenses putting together success for a time. It's not as radical as it used to be, but can be as potent.
This would be an intriguing fit for Newton specifically, with a lot of shotgun and flexibility. There would be a worry about structure, however. Since so many versions have adapted, there's not a common philosophy - there are versions of the spread that derive from the Wing-T, for instance. Without a lot of commonality, this offense is tofu - it'll take on the flavor of what you put with it. And that probably isn't enough structure for a young quarterback. Since various offenses are changing to fit the college game now, it's not necessary to tailor something new to do the same thing.
So those are the basic options, and how I view their adaptation. I'd really prefer to stay with the same offensive staff, and if not, the same system.