It may be too hard to match the wood colors, but you should think about doing plantation shutters. You can hang the shutters up against the window. That way you have light control and can still sit in the seat. Also, plantation shutters can be done in many shapes, including the curve on top of the window.

From the looks of the height of the chair in front, it looks a little too high to be used as a window seat. I would suggest a simple custom fitted roman shade made to match the arch of the window. I would place it as close to the glass of the window as possible and not at the alcove where the bamboo blind is currently placed. This will take full advantage of the shelving/display space that this feature has to offer.

I'd put a curtain rod at ceiling height, and extend it at least a foot on either side of the window, then hang floor-to-ceiling drapes.

We had this exact same problem at the school where I studied architecture. The solution found was bottom-up blinds installed close to the plane of the windows. It can't go up to the very top of the window, but it'll cover most of it.

Try a window film? You can leave the window bare, that way.

If you do a google, 3M and Gilafilms pop up with products (cutting glare and heat) that might be just perfect for your situation.

Raised panel Shutters, but not plantation shutters. The panels could mimic the wood around the sill. If they are a little bit too pricey, I'd second the roman shades (top down). It looks like the ceiling has been lowered a bit, that's why I wouldn't do the floor-to-ceiling drapes.

I would still put in a cushion and drapes and make it a window seat. Having to climb up into it would make it seem like a fort (I guess I will always be a kid at heart). Then if you don't have a cat, get one. Prime kittie napping spot.

You could make a roman shade out of any fabric you like, and put a backing on it that would block out the sun during the afternoon. Maybe even put them 3/4 of the way up to keep some natural light?

The heater is totally going to ruin that space for a window seat. That said it is the perfect space for one.
In regards to covering it, I say tension rod partway up the window flush to the glass with translucent curtains to let in diffused light.

You need to cover up that edge at the top, whatever you do.

FYI: I also have arched windows, and shutters will NOT work in a deeply recessed windows. The sides of the sill are lower than the tallest part of the shutters, which will get stuck half-way.

what jenPDX said: window film

With the bottom heat from the radiator and light it sounds ideal for plants. Tall plants, trailing vines, etc. to partially block light but such a pleasure to have during the cold dreary months especially.

Do you have to keep that iron grate outside the window? If you do, consider growing a vine that will create a pretty screen for the light. Ask at a private nursery -- not the garden center at a chain store -- for advice on what will grow in your area and maybe keep its leaves in the winter.

Consider the heating situation first, as a separate issue. Do you have to use that heater? I find that it's more energy efficient to heat only my space. I use one of those rolling radiator-looking sealed-oil heaters and keep it near me. If you don't use that heater, the wood will remain in better condition (you are going to restore the wood - please don't paint it! And allow me to introduce you to the wonders of Howard Furniture Products.) and you will be able to use it as a windowseat. How could you not?

To use it as a windowseat, get one of those adorable chairs (meant for use in a library) that double as steps. Or find some unfinished boxes and configure them like a tansu and finish or decorate them to your heart's content. Bonus: extra room for books.

The secret to controlling extreme temperature is layers of fabric. It isn't hard to bend wood to match that gentle curve at the top of your window. DIY it or get some help and make a curtain rod that preserves that beautiful curve. Hang some wonderful sheers. These will screen the light.

Next step: There are drapery rods you install directly into the ceiling (I have these.) The low ceiling won't matter.
Install two of these, one in front of the other.

On the rod closest to the opening, hang another, slightly more opaque, set of curtains, sort of halfway between sheers and regular curtains.

On the final rod, hang some nice heavy drapes, such as a good weight dupioni or velvet or corduroy or damask -- something with a little thickness to it. Velvet is best -- very good at insulating against both heat and light.

You will find that with this three-layer arrangement of curtains/drapes, you'll be able to control the heat/light situation as you please throughout the year.

And most definitely make the seat cushy with pillows and at least one cat. I wouldn't put any shelves in. The wonderful wood is much too special to hide.

As the perfect final touch, if you design the tansu table cleverly, you can turn one part of it upside down and make it fit the other part so that the final version is a square -- think of Tetris shapes -- and keep a tray with a pretty teaset on top when you're not using it for stairs.

Good luck! Send pics of the final product!

P.S. If you decide not to use the heater, could you possible remove it? It looks as though the front of the heater consists of two matching door. Wouldn't it be great to have all that storage space under the seat! Might be a good place to keep your space-heater when the weather's warm.

lose the blinds, improve the finish of the "seat" surface, perhaps add some simple pretty sheers to counter your heat problem, and lovely cushions for lounging.

I'd go with shutters. I have wood blinds in my office space that were here when I moved in. The aren't pretty, but they work well to block direct southern sun in the summer time. They can be angled to let in as much light as I like while blocking direct light that makes it impossible to see my computer screen. If you could find some sort of shutter, you could get the same effect with better aesthetics for that window. A shade of any type will make it quite dark in that room, but perhaps you wouldn't mind that. Plants would be beautiful there, but they either wouldn't give you the filter you need or they'd make it dark all the time.

Deeply recessed windows are made to work with shutters. Town homes from the 1800's in Manhattan sometimes have their original shutters still in place on their deeply recessed windows. The shutters fold flush against the side wall of the window (perpendicular to the window) and are solid wood and detailed to look like old fashioned paneling (as in library, not 60's basement). This window may have originally had these very same shutters.

i vote with michelle of montreal. a simple white bottom up solar blind, roman blind or roller deep in the niche close to the plane of the window.

i'd be tempted to sit in it despite the height, so i might add a cushion. if not, at least a nice plant.

i'd also like to paint that pink/beige wall....that color is institutional.

PLANTS SHEER PANEL MEDALLIONS TIEBACKS

I'm with tikitimbo's suggestion: fill the alcove up with plants! At the very least, hang just 1 large trailing plant from above... and to help diffuse the remaining light, you could hang a sheer from drapery medallions over the arch (tie-back on side).

Here's a photo of medallions in use: http://flickr.com/photos/11080268@N08/1225869618 </a href> (of course you don't have to use that dreadful fabric). Here's another pic: http://bmwblinds.com/bmw_draperies_valances_files/image008.jpg </a href>

Drapery Medallions & Tiebacks: http://www.windowwearbydesign.com/drapery_hardware/decorative_holdbacks_medallions.html </a href>
You would likely need to install the medallions directly onto the arch molding since there's no clearance above the window.

The shade needs to be against the window, not the frame, the bamboo shade is in the wrong place that's why it looks so bad.

Keep it simple.
I like the look of wooden blinds aganist the inside, not where the blinds are now. Or wooden shudders aganist the window.