No seriously these are beautiful

Could I see Animal Crossing and either of the Zelda games? Please? 

I'd say it depends on which resolution you want to play at. Any modern CPU is reasonably capable to play DS games at native res. This version of the emulator (which is not an official version and isn't updated) has given some people with i7's (but low clocked ones < 3.0ghz) trouble, while it runs fine for the more popular games on my overclocked (4.4ghz) Pentium G3258. You can play the games at 2x, 3x, and 4x res as well.Also I'd expect a discrete card with at least 1gb of GDDR3 (or better) Ram would be enough for the GPU end of things if you want to play at 4x res. Some games work better than others for me. I noticed slow-down and sound issue in Goldeneye when I increased the resolution, but I might be able to fix that if I tweaked settings. Also I noticed that neither my CPU or GPU was getting loaded to max, so it might just be a coding problem in the emulator. If you want more information,  google "HD DS Emulator." It is one of the first results. I won't post it here because of forum rules. 

It bothers me that companies don't bother upscaling classic game on their virtual shop (eshop/psn/live) on powerful consoles. I think Banjo-Kazooie on Xbox360 did, but I know that a lot of PS1 classic I own on PS3 are direct port without upscaling. It should be easy no?

Well these games are being natively rendered at these resolutions (not upscaled), but yeah if random people with no documentation can make an emulator that renders games at higher resolutions on open platforms, it shouldn't be too hard for a team to be dedicated at these big named companies to achieve something better on a closed platform. I think their biggest concern is accuracy. They don't want to make a general emulator because they want all games to be 100% accurate; however, the accuracy in these emulators is fine enough for consumers in my opinion, and these big name developers can accomplish much more, especially considering they have documentation of both hardware and know how it works. Furthermore, they can have an configuration file that sets the emulator to the proper settings for each individual game. There is no reason why the PS4 shouldn't be able to emulate PS1 games at 1080p 4 xAA for example that is reasonably accurate. Possibly they think the marginal costs are too high for a measly marginal benefit. 

WOW!!!!!!! AMAZING!

Simply beautiful. A true eye opener to anyone who might think resolution bumps do not make a big time difference.

The increase in quality this Zelda title experiences is nothing short of astonishing.

Wow. Some of these games (Zelda, DQ IX) would be impressive even for Vita games! Shocking that these are DS games. I want that DQ IX ported to my Vita!

Dang, those look really good. It's amazing to see the improvement just from raising the resolution.

Not sure if serious...