Suicide: the act of killing oneself.
If you find your alternate self and kill it, but you are still alive, then no. You have not killed yourself.
Also, I like what Tsuki said a few posts back,
Tsuki wrote: "Well it could be argued that Suicide is actually "taking your own life." In that sense, seeing as though your alter-ego is not actually consciously a part of you, it is not actually you, which would make killing them simply an act of homicide."
In order for it to be a suicide, you have to show intent. When someone falls asleep at the wheel and drives off of a bridge, it is not labeled as suicide, even though they have killed themselves, because the intent is not there.
Even in the case of murder, which I'm only bringing up to support my theory of intent. You can only be convicted of murder if the prosecution can show your intent to kill (among other things, of course.)
In the case of killing your alternate self, the intent is to kill that person, not yourself.
My thoughts!
Lilaiy