Now here is the trap. :) "Engemet" is technically okay but it is a lower class / obsolete form. You can meet this in old folk song lyrics (for example, the most famous is "Tavaszi szél vizet áraszt") and in the country here and there. But in general we use "engem" and "téged" instead of °"engemet" and °"tégedet", dropping that da__ed accusative. This is an idiomatic thing, don't even ask the reason, it makes easier to learn it :D
Your example "Te keresel?" for "you're looking for me?" is right, but not in every situation. We tend to drop those parts of the speech that is obvious for a reason. Some examples:
- You hear someone calling your name, but you're not sure who's calling. You may ask your friend "Te keresel?" and it will be obvious that it means "Te keresel engem?".
- You find your computer with Google open, though you left it for a moment with a spreadsheet on screen. You may ask your friend whether he used the search engine by asking: "te keresel?" Here it will be irrelevant whether he is searching for you (e.g. on facebook) or he is searching for the nineteenth president of the U.S.
- You're talking with a friend about unemployment and he tells you that almost nobody could work in his family. You may ask him whether he has an income by asking: "te keresel?"
All these situations make it very clear what is the question about, and you don't have to mark it.
So, did it feel suitable? :P I feel like even "te guglizol?" or as one of my professors said, "guglázol" would fit better. That "on-demand" verb doesn't have a mandatory argument at least. A verb that can describe an activity rather than a goal oriented action.
But anyways, these aren't facts, these are personal experiences with the language that can result into normatives, kinda. I wouldn't recommend using "keresel" this way, although it's not objectively wrong. It just feels very lazy. :P