I've had this difficulty as well. Of the many story ideas, practically in the hundreds, and the numerous attempts at writing them down, only a handful have ever been completed. Most of these were for school assignments, such as in my playwriting class. A few I've done just for myself.
Sometimes, to keep focused on an idea, you have to write it down. Write down anything you can come up with. Character biographies, world descriptions, plot outlines, lines of dialogue. Anything and everything. Having something physical in front of you can keep the idea concrete in your mind, and keeps it from just being an intangible idea that can easily flitter away. Drawing out the ideas also work. Talking over the ideas with friends and other writers can also keep you interested in the idea, and listening to how others react and to their opinions on the idea could help you improve upon it.
When another idea comes, don't let it go, but don't let it overshadow you're previous idea. I find that I can juggle around with 2 or 3 ideas at a time. Play with one in the morning, than switch over to the other in the afternoon, and then deal with another at night, or any way you want to do it. Some times, staying focused on a single idea without a break can cause you to get bored of the idea and can cause you to lose interest. So playing with multiple ideas is actually a good way to keep you excited about them all.
But what really helps is setting a deadline for yourself. Mark the date on a calendar and tell some friends so that they can help keep you on track. Choose a time and a day when you can remove all distractions for a couple hours, shut down the internet, turn off the cell phone, and put away the games, sit down in front of your computer or notebook and just write. Don't get bogged down by details and write whatever feels write for you. If you find the right idea that excites you, it won't leave you, and you will want to get it finished, and the words would just flow.
It can take multiple attempts to finally get it right. It's like running a marathon. Nobody expects you to finish the first time around. Practice practice practice.
Yep, had that happen, too. I've got a number of partially-completed stories (and only-just-started stories) that have been abandoned in favour of other ideas. Sometimes when it gets down to the "I'll need to research this bit to get it right" stage, sometimes when I realise I'd have to do years of research or actually take a world tour and visit some of the places, sometimes when it just needs a tidy up and I can't be arsed re-reading it over and over. And sometimes I just get a new idea that fires me up more.
Do you ever go back to half-completed ideas and continue them later? (Not by forcing yourself, I mean, but just spontaneously.) If so, then I wouldn't worry about it. I have probably several hundred half-finished stories around the place, and when I'm inspired to work on a particular one, I work on it. I've finished (but not published) 6 full-length novels by now, so not sticking long with any one doesn't mean you'll never finish anything.
If I'm asking for advice on a story idea, don't tell me it can't be done.

So, my problem requires a little exposition first.
When I was a child I made stories to entertain myself. They were nonsensical garbage when I look back on them now, but were fun at the time. I enjoyed it, and wanted to continue doing it. However, as I grow older I find it harder to make stories. I must adhere to a higher quality now, which means any given story would typically need tons of research and thought.
The problem is that when I feel in the mood for a particular story, say Fantasy in an Asian based culture, I might make up a few things. But within a few weeks I come up with something else. I lose interest and focus before my story has moved passed the bare-bones idea stage. This has happened more and more frequently, and now I can not come up with anything. The conclusion is that I have not made a true story in years. It does not help that nowadays I have many more distractions, as I have a job and what not.
I ask if anyone else has come under this particular lack-of-focus problem, and if there are any suggested remedies?