I have been looking everywhere for the answer to this question. I knew it wouldn't be crossplatform with PC/Mac, obviously, as the PS3 version was not and is not, but I wanted to know if I could play Diablo 3 on my PS4 with my PS3 friends. So without further ado, the answer is a resounding NO.
I purchased the PS4 version and was unable to join games with PS3 friends. I knew it was unlikely this would work, but I had to try. Blizzard had been unwilling to respond about this feature, probably because they weren't sure if they would eventually implement it, but that they couldn't just be upfront about it in regards to the release really annoys me. Would it have hurt them so much to just say it?
It is clear that many games have the ability to be played between PS3, Xbox 360, PS4, and Xbox One. Need for Speed Rivals was designed to do it, according to the developers. Destiny was definitely going to do it until they recently decided not to. Diablo 3 was a perfect candidate for this kind of play and Blizzard, just like everyone else, totally let us down. Need for Speed didn't seem to give a specific reason for why they changed their minds. Destiny claims it would ruin PvP and that newer generations would have an advantage, which I think is bullshit. If you look at the difference in quality between the games it's negligible and if you were still worried you could make it so that PvE could still have the ability to run cross platform. But Diablo isn't even so much a PvP game like these other examples. And even when it is the computing power isn't going to make the difference.
Some people have mentioned that the difference in computing power is the cause for this separation. That's also bullshit. Computer games have had scaling graphic options for decades and they were still able to play together. So you build it to run with the lowest necessary settings and BAM, you're good to go.
Needless to say, I am seriously disappointed in Blizzard and pretty much every other modern gaming company. So many times it has been mentioned that games will be crossplatform or that we will have that capability and so many times, as we reach the release of a new game, we have been let down. I'll at least give Blizzard the credit that they never promised Diablo 3 would work in this fashion, but they never really denied it either. Or, at least, it wasn't easy to find an answer.