Thanks for the fantastic pictures, those really help!
Unfortunately most replacement shower hoses have 1/2" pipe thread female ends, yours has male ends. By any chance is there a brand name anywhere on the fixture? I've looked at a couple of places and haven't found one with male ends yet. If you have a plumbing supply house nearby, you might try there.
The other alternative would be to "make" your own hose by using a standard female ended hose and then putting short nipples in each end to basically change the female end to a male end. Although Home Depot doesn't sell them, you can buy short chrome nipples for a better color match. Stainless would probably also work because you won't see much of the nipple either.
The brand of faucet is Neoperl. I don't know if you know that brand.
By the way, I can't take your alternative choice because the extension hose is supposed to be under the bathtub and it comes out when I use. So the joint should not be wider than 3 fourth inch.
Thank you for your kindness anyway and if you find any further information with the brand, please let me know!
It appears that NeoPerl is a Swiss company that seems to sell mostly in Europe but they do have an American office. I would suggest contacting them directly for replacement parts:
Yours is a great example of why to buy big name "American" plumbing fixtures. It's not that they are better or that you should "buy American" but rather when it comes time for service parts are just so much easier to find. I can walk into just about any home center or hardware store and find a Moen #1222 cartridge.
If one is looking for a "premium" fixture, Moen, Kohler, and Delta all offer them. You just won't find them at your local home center.