Hi,
hi C tuning is fine - though don't try and use a normal 5 string set! You can either get a high C set, or a four string set with the thin one off a six string set...
The only possible problems will be with the strings moving about in the nut slots which which will be cut for the thicker strings, but I've not actually met anyone who's struggled with this.
There are quite a few very fine bassists who tune like this - the most famous I guess is Steve Swallow, and Tony Levin does a fair bit of it on his latest (and fabulous) solo album 'The Waters Of Eden'...
cheers
Steve
www.steve-lawson.co.uk