does anybody know what this is?

I'm having my kitchen remodeled and I am trying to figure out what this is (see attached image).

It appears to be a vent of some kind (there was a perforated cover that screwed onto the end which isn't shown in the picture). Nobody can figure out what it is (including the plumber). It is located on the wall an inch or two above the counter top above where the dishwasher is.

One time when the drain pipe backed up, sludge water came out of it if that gives any kind of clue.

Sounds like an island vent to me. When a vent cannot continue up due to a window or when you install a sink on an island, you would have to turn the vent back down and tie it in elsewhere. It's a bit complicated to explain but it appears to be the case here. The pipe coming out of the wall would be the cleanout.

Sorry, I just saw that it is above the dishwasher. Is there a drain line for the dishwasher? It could be an air gap.

That copper pipe you see coming off of it leads towards the dishwasher but the drain line for the dishwasher is hooked into the drain under the sink.

looks like an island vent

This nipple is sticking out of the wall is above the sink counter top?
I have seen this before in La Jolla, Ca. just last year.

It is an air gap for the dishwasher built into the wall.

Often custom home builder try to eliminate the common unappealing eyesores like air gaps.