I was looking for a C70 earlier this year (before a free 244 landed in my lap).

I wasn't considering one so new as a 2004, so I don't know how they differ. The C70 is an outlier in the Volvo stable, so many of the problems are related to things that are unlike the S/V70. I read that part of the design was contracted out to Tom Walkinshaw Racing (TWR) in England. Most of what I gathered about them centered around problems with the rear window regulators amd the power conv. top. The windows have no frames, so presumably the regulator design is different than the "regular" models. So a new design coupled with the fact that the windows get used a whole lot means that they are known to fail. The convertible tops give people fits because... well because it's a power convertible top and like anything else, it will eventually get messed up or fail. There are diagnostic codes just for the top. Sometimes they will get set off even though there is nothing wrong.

The other common complaint was about "scuttle shake". The convertible chassis is inherently weaker (less stiff) than a coupe, so some annoying movements are observed during cornering, over big bumps and heavy acceleration. There are aftermarket braces that reduce this. My answer to that was to try for a LPT model with less power and smaller wheels instead of the HPT one with gobs of power, stiff suspension and 17" wheels with no compliance.

Again, this is all just stuff I've read and now regurgitated from memory, YMMV. After doing all the research I was a little put off, but then I realized that most (if not all) of the problems are the exact same problems that every other power top convertible cars have. Just find one you like, make extra sure everything works, look for service history, and enjoy.

Best of luck,
Will
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850 / 240 / Mini