If your lacking storage space, I would suggest closing the space off and making a storage nook.
See link below.

You could put some pictures or figurines up there. That's what I would do.

You could add a ladder to the space and make it a cool mini loft for books or some colorful art! Bartels offers custom ladders to make this a beautiful focal point rather than an awkward nook.

If it were my space, I would measure it, add the same size basket with handle all the way across, so I could store out of season (coats in summer / shorts and bathing suits in winter) clothing out of the way. It could be really cute to do this with the door and the baskets done in the same accent color - for instance, you can buy cardboard bankers boxes at the office supply - they come with tops - the discount version is to paint these . . a pack of 12 is $40 http://www.officedepot.com/a/products/199570/Bankers-Box-StorFile-Basic-Strength-60percent/
Paint the cardboard with craft acrylic paint in your accent color. If you like the funkiness of painting the door too - get a quart and accent the door too . . .

Decorative: If you don't mind dusting, some interesting artifacts, clay bowls, wooden masks, etc.
Storage: add some large wicker baskets (see size in 2nd picture) and store seasonal clothes
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Yes, this can be used for storage or display. I have seen large primitives and collectibles displayed on ledges like these. However, dusting anything that is placed there will be a job. Short of building closed storage, you could opt for storage bins with lids - try West Elm, Crate and Barrel, and the like.

Do you collect wine or books? - might be an opportunity to do something different.

I would not want to put anything decorative up there. Just ugly dust catchers. Is this a new house? was it in the design? At this point I would hire a cabinet maker (not high priced) to build cupboards w/ doors for suitcase storage or for out of season clothing. Be aware it looks like your ceiling is vaulted so the tops of the doors will not be full height in order to be able to open without hitting your ceiling. It's either that or have the wall extended to line up with the closet depth & have lost storage. I agree it is not pleasant too look at.

I think it could look great like in the pics, you've got it, so make it a feature, you could even add soft lighting, I'm sure dusting it every few weeks is manageable, make it fun

How about leaning, framed pictures?

These are wonderful ideas! The amount of dust up there is disgusting. The surface is textured for some reason too. I had only gotten as far to think to put contact paper down to make dusting easier.