That red -lit building would be a hospital. Behind it on top of the hill stands Hallgrímskirkja.
Upon arriving to Reykjavík, one of the first things you'll notice is a huge bubble-shaped building on which seems to be a hilltop in the middle of nowhere. There's a reason why you can see things in the middle of nowhere from the Capital City, and that is mostly lack of trees, but I digress.
This building is called Perlan or, The Pearl.
The Reykjavík airport at night. Not to be confused with Keflavík airport where all the international flights land and take off.
During the day the building grabs your eye because it looks so unusual, and at night you'd likely see the lights going around the top of it like in a lighthouse. Inside the building hosts f.ex. a wax doll museum (a real must-see!), a fancy restaurant on the top and a wonderful cafe. The restaurant, being far out of my price range, I haven't yet tried. The cafe and its selection of ice creams has been duly sampled and well... ice cream in Iceland is always heavenly, but of all the ice cream the one served at Perlan's cafe must be near the very top!
Here's my dearest Alma Mater as seen from Perlan. It's actually much further away from Perlan than this photo might make it seem.
These photos have been taken from the viewing platform that goes around the building near its top. The only thing above it would be the restaurant, and if I'm not mistaken the restaurant is revolving so they'll get the easiest 360° view of all from up there. However, the platform is open until quite late and worth a visit any time of the day, really.
Here's a 180° view from Perlan. Clicking it open in a new tab will make it huge.