Different airlines handle this issue differently. Some airlines will allow you to carry on w/ advance notice. Some will require you to purchase a second seat. In the past, the second seat option has been offered at a discount. However, last time I flew, the airline I flew with no longer offered the discount. The basic message here is contact the airlines, tell them you are a musician carrying a valuable instrument who would like to know what your options are for the instrument. Be prepared with its dimensions and, if they do tell you that you will be able to carry the instrument on the plane, be sure to find out if there are any special procedures required during check in and security-clearance with the instrument (many security stations these days have mandatory limits for size of carry on items).
If you decided to ship in a flight case, be sure to get a high quality flight case made specifically for your instrument. The folks at Alembic can help to hook you up with this. This is very important, as I know that other Alembic owners have had woes shipping instruments in regular hardshell cases. Can't remember who, but I recall one of the club members telling a woe-full story of receiving their instrument with the headstock punched through the end of the hardshell case :-(
Also, if you decide to check it, you should still consult the airline to determine what level of liability coverage they have. When I flew, the value of my Series II FAR exceeded the liability limit of the airline's baggage insurance. You may wish to obtain full replacement insurance through a musical instrument policy like those offered by Clarion, etc.
I have recently flown using a custom flight case that I checked and it worked out well. However, no matter how good a case you buy, remember that once it is out of your hands, all you can do is sit back and breath deeply until you pick it up on the other side :-)
Simon