The NAD VISO HP50 for on the go. See:
the HD 600 or 650 for home with a decent amp (the affordable O2 will work fine). ...
Although they are both very fine cans, neither the HD 558/598 nor Q 701 have the type or amount of bass you desire.
I have not heard them yet, but I'll bet someone here will recommend the SoundMAGIC HP100.
I recommend the NAD VISO HP50. See:
I don't have any cost of manufacturing or any other inside information. My guess is that the 32 Ohm pair compete with cans designed for portable use. That means their competition is overpriced cans like Beats.
The DT 770 (in whatever flavor) is not a closed version of the DT 880. The DT 770 has a "V" shaped sound signature with very recessed mids. It is also a very large headphone. The more efficient versions of these cans (80 & 32 Ohms respectively) are not as good as the 250 ohms and up versions. See: http://www.innerfidelity.com/content/comparison-beyerdynamic-dt-880-32-ohm-dt-880-250-ohm-and-dt-880-600-ohm-headphones
Only if you get amp, should you consider buying a 250 Ohm headphone--like the DT 880 you are considering. Unless you computer has a sound card (not on-board sound or a chip) and that sound card has a built in amp that will properly drive 250 Ohm headphones, it may do worse (and will certainly little better) than your iPhone.
At the top end of your budget, you can currently reach the UE 6000 and CAL! However, these are not bass heavy headphones. So they may or may not be in line with the sound signature you seek.
At your budget, there are few quality options. Especially at $50, you will get more for your money with IEM's.
In the U.S. the two best cans that can currently be had at that price are the UE 6000 and the CAL! However, they won't satisfy bassheads of any stripe.
If these suggestion don't help, maybe you can tell us your country--as well as what will be feeding and powering these cans.