Almost all heating systems (home designs) have hot air output right under windows. One reason for this, or coincidence, is the reduction of moisture/mold/mildew in the house. Blowing hot air right near a cold moist window reduces the moisture and condensation. This reduces the mold/mildew growing and humidifies the house more (instead of leaving all the water on the window, it gets redirected and moved into the air). Blowing hot air on the moist window will increase the humidity in the room.

The window is also the coldest part of the home, usually. If you blow hot air on to the cold parts of the room, it could keep the temperature more even throughout the room. However, this could also cause a cold draft. Heating the cold window might move the cold air around too much, and cause drafts or even small cold winds in the room, making the whole "heating the cold window to improve comfort" theory a failure.

If you do not sit near a window, it may be wiser to blow air slightly away from the window with deflectors. This way you are directing the heat to where you are sitting, leaving the cold air temporarily aside. Eventually, the cold air is going to catch up with the heat - but since it is in ongoing battle to heat a room, this redirecting of the heat would temporarily give you a warmer room in one section (where you are sitting, hopefully).

Redirecting the heat could leave the part where you are not sitting colder (near the window, leaving a cold cloud of air sit there, but in an ongoing battle of course). This is similar to why you wouldn't try to heat the entire house if you just live in one room. Heating the entire house is not going to save you energy costs either. Closing doors to rooms that are not used does save costs, because it takes longer for the furnace to notice that the room that is blocked off is cold. Similarily, I theorize that if you direct heat away from the cold windows, it may take longer for the furnace to realize that there are cold spots in the house, and save some energy costs. Since we don't generally spend much time near the windows in the winter time, heating the windows may not help save energy costs - it may just help humidify the house.

A window may be somewhat similar to a radiator in a car: you need a fan in your radiator at stoplights so that your engine does not overheat. Eventually the heat in the radiator will go away if you have no fan, but it will take much longer. Similarily, a cold window will take longer to heat than if you just heat away from the window. If your window is away from the window, eventually, your heater will catch up with the window and start heating it when the air arrives there. But if your heater is directly under the window, it will take longer for the room to heat up since it first has to heat up this succulent reverse radiator. (reverse radiator since the window is cold and sucking out heat). Your window is not like a radiator in that you WANT to move air around it and shuffle cold air around - that cold air is better left in a cloud of it's own, away from you. But then we have the disadvantages of mold/mildew growing around windows if we do not place the heaters directly below them.

If you sit near a window, or you stand/sit in all sorts of places in the room, it's probably more comfortable in the end to heat the entire room anyway, so maybe heating under the window isn't as silly is you might think. However, this being said, it would also be wise to consider insulating the window better (plastic windows, warmer drapes, etc.).

You have to admit that heating a cold window in any way is still inefficient. It's just like trying to justify that heating an ice cube is some how going to help the temperature in the room, even if you could avoid that ice cube and delay being around it, letting the ice cube sit in a designated cold spot. Better rid the ice cube, avoid it, ignore it, move away from it, keep it segregated, or some how block/insulate it from robbing you. This is all about direction, blocking, relaying, delaying. Heat is not as simple as just insulation, radiation, and uniformity. Heat is more complicated, and includes all sorts of factors such as the direction in which you aim the heat, the segregation of the heat (hot spots/cold spots, uniform hotspots, non-uniform hot spots, etc.).

If you can keep the window area warm and dry, then you increase the humidity in the house without causing mould and wood rot. Hot air rises up the wall and window, and meets the cold air that is falling from the top of the window. The two hot/cold air streams join together and flow into the room at about face height or higher. The problem is that this can cause drafts. If the flow is weak enough, it may not cause a draft and may be a way to increase comfort in a room (but there is still heat going out the window too, let's not forget our points made in disadvantages).

Basements are usually cold, even in the winter, even though supposedly they are more insulted. The other factors, such as heat rising from all the furnace ducts in the house, seem always to heat the main level or higher levels of the house. Not sure about high rise buildings, the top floors may or may not be hotter than the main floors.