Is anti-aliasing still relevant in a high resolution display?

Yes, anti-aliasing is still very relevant. The reason is that the human eye can see subpixel movements on most displays. You'll often see pixels per degree as the metric used and reference 60 PPD for "retina" displays. The human eye at the center of vision has hyperacuity of small pixel movements that go past 400 PPD.

1. Do you need anti-aliasing at higher resolution?
2. Does 1080p need anti-aliasing?
3. Do you need AA at high resolution?
4. Do we really need anti-aliasing?

Do you need anti-aliasing at higher resolution?

It depends on your purpose. If you are just looking at a monitor with the average human eye from an average distance, then somewhere around 200-300 ppi will make individual pixels nearly in-distinct and so anti-aliasing wouldn't make much difference (but still some difference depending on how you use the display).

Does 1080p need anti-aliasing?

AA still has an effect even if your resolution is maxed out, because of jagged edges. At 1080p you will still see 1080p jagged edges of a straight diagonal line.

Do you need AA at high resolution?

With high resolution you're still going to want some sort of transparency or alpha antialiasing. Until the size of textures themselves become insane we're always going to want AA. Shadows and HDR lighting are also a couple of the biggest causes of obvious aliasing these days.

Do we really need anti-aliasing?

Among all of the graphics settings you need to tweak in PC games, anti-aliasing is one of the most important. It can tank your performance or make your game look terrible if harnessed incorrectly. But with a little guidance on what anti-aliasing settings to use, you can optimize any game in seconds.