Can you create sort of a bump-out or false wall that is bigger than the window and put LED light strips in there to light it up? Just make sure that somehow you have access to it from the other side to get to the electrical. What's being added to the other side? This bump-out could be hidden in a bookcase, or something depending on what type of room you are building.

If your concern re: light is to get sunlight into the bathroom, you could add a sun tunnel.

If it is more so that the window itself just doesn't look dark, you could either do some sort of lighting behind it, or make it a design feature on the other side as well instead of covering it. Obviously it won't be as brightly lit being enclosed in a new room, but the natural light from that room would at least keep it from looking dark. Its not entirely uncommon, although you'll need to get creative with how to make it work in whatever that other room is.

the part of the addition behind the window will be a hallway

Light it with a spotlight in the hall. Use a halogen bulb to get the broadest spectrum of light. Connect it t the same switch as the lights for the shower. Piece of cake.

Don't use LEDs for this; you need a clear, warm point source of light for best effect.

And if the sun shines in now during your normal shower time, aim the light from the same direction.

you can add led light strips to re create natural light very easily. they come in different hues from warm to cool lighting. this is a false window we created for a client in a basement sewing room. the window was existing but it was used as access to a crawl space so it had no natural light. our client didn't won't to be in a room without a window so we gave her one. the light strip is on a dimmer so you could control the brightness. If your addition will have a hallway behind it, you will have access to be able to get the lights behind your oval window. you will have to figure out with your electrician where to put the switch, but I imagine you will have a electrician involved with the remodel.

OK I appreciate the ideas for the light. If I don't care about lighting the window, does anyone have an idea for design? example, frosting the window so you cant see the studs behind it in the addition? or just making the background black?

If you build a frame around it with studs as it if were a larger window (making headers, etc.) instead of studs 16" on center you wouldn't see studs behind it, just the back of the addition's drywall which you could paint white in that area. I do like the back lighting suggestions already given. You can buy spray cans of frosting paint if you just want to paint the back of the glass.

You could have a mirror cut to fit, which you could just call a fancy shaving mirror. ;) Alternatively, you could have a PVC shutter made for it, either solid or slatted.

I was wondering what the mirror would look like. just didn't know if that was weird to have! thanks!