Stephan (avtar name) did this - he had a perfect round made to strict proportions. You might want to read his recent threads and get in touch with the respective people (LINK). The "production chain" is already in place
You might also want to read Garry's (avatar name "CutNut") account about the recent conference on Diamond Cut LINK).
... and definitely inspect GIA's exaustive list of exiting diamond cuts (LINK) - it makes good inspiration and hint on what is already there and what is feasible (since the propoerties of the material and the purpose pose constraint that migt not come as straightforward to you, depending on your background - which I ignore).
It is just a small leap of faits (and use of already traditional technology) to get a new shape. This does not stryke me as an economical solution or a fast one, but why not!
Actually, there are so many shapes and versions out there, that you could well find something 'close' and let a cutter develop your ideas to meet constraints needed for durability and brilliance.
The cost and duration of this will certainly depend on what exactly you decide on.
And... have you heard of the "flower cuts" ? for example ?...
Outside of a round brilliant, I would never say never, but I would venture a highly unlikely.
If you're thinking of asking a diamond cutter to design a unique facet pattern for you, find the rough and execute the cut, my guess is this would be highly doubtful/impractical.
If you go with a round, the expense, time to completion, and success of the final outcome will vary depending on if you start with a piece of rough or have an existing stone recut.
Vendors of Infinity Diamonds (
), will consider custom proposals from clients, but the approach would be taylored to each individual situation.
"I would love something really special and unique"
Have you considered having a gemstone otheer than diamond custom cut say a quality sapphire or ruby?
I have designed some custom cuts that someday when I get some money ahead I will have cut.
It is a ton of fun! and the end result will be well worth it :}
There are several people that could make that a reality :}
Check out the colored stone forum.