A very interesting question. Rudolph was apparently more liberal than his father, so having him survive could have had interesting butterflies. Problem is the longevity of his father, Franz Joseph (who only died in 1916): by the time Rudolph would get the Austro-Hungarian throne, it might be too late to save Austria-Hungary.
Well, come to think of it, that wouldn't make Franz Ferdinand the heir, since Rudolph would still be alive, so the OTL 1914 Sarajevo assassination might not happen, butterflying OTL World War I. Yet, if we assume Franz Ferdinand was targeted because he was the heir, then Rudolph could be the target of assassination attempts.
Who knows?
Rudolph didn't have a happy life, and I think he did show he was depressed because of his unhappy marriage. So, the possibility he did commit suicide is there.
However, one has to admit there are troubling facts. Rumors said there were traces of a fight in Rudolph's chamber. Another thing is that, when his corpse was shown, Rudolph was wearing gloves: was it to hide scars and bruises proving he did fight for his life? Lastly, there is the fact that the last Austro-Hungarian Empress, Zita (wife of Charles I & IV), said that Rudolph hadn't commited suicide. The possibility Rudolph was murdered is also there.
But if he was murdered, by whom? The most common theory is that it was a French plot. Another theory I heard is that it would have been ordered by the Wilhelm II of Germany, as Rudolph could have shown himself a Francophile. There is also a theory he might have been murdered by Austrians because he was pro-Hungarian.
For my part, I can't say as I wasn't there and didn't witness anything. Thus, while I do admit the possibility he was murdered is there, I stick to the suicide theory until it is proven wrong.