I have the 440s. I like them very much, but they are not bass-heavy headphones in any way. That being said, bass is still there, just with a more 'quality over quantity' approach; bass detail and extension are both good, not to mention across the spectrum. Even though I listen to mostly modern alternative rock, I don't consider myself a bass lover, and so I'm happy with them. But for hip-hop and gaming, I don't think they'll quite deliver; Shure calls them 'professional studio headphones' after all.
I haven't heard the 840s myself, but people say they're warmer sounding than the 440s. They might suit you better, but that's to be taken with a grain of salt as again, I haven't heard the 840s before. The D1001s mentioned by xnor might also be more suitable; I bought my 440s because they were collapsible unlike the D1001s, but you might not need that function.