Hi,
If you take good measures to protect your property there's a much less chance of theft etc - don't take Cheers' little scaremongering there to heart !
The last time I checked into this sort of thing was for both ways, for me exporting a car from the UK to the US and bringing a car from the US to the UK. For the West Coast I was quoted around £1700 and the same price for both options above with a 21 day sailing time. A container with a car inside is of course quite large so a motorbike being much smaller could be put into a solid crate and then perhaps put into the container with your other belongings as well. Or, you might find a deal from one of the shippers where you can be quoted to bring the bike across and it'll be in a shared container that's coming across anyway.
Do check on the legalities of importing a bike, it's fairly straightforward though but it is something you need to research prior to bringing it in.
Cheers, DtM! West London & Slough UK!