why not "penceresi"?
In this case, you have to use the dative (which is always used when talking about motion towards a place). penceresi would just mean "his/her/its window" :)
I must had missed that, thanks!! That makes sense now! :)
What about pencereyi? Can ı use it?
The postposition "doğru" takes the dative case. It never takes the accusative case :)