Ven a buscarme.

English Translation

Come get me.

Does this not translate to “come and look for me?”

I believe Come look for me and Come get me are both correct.

I think “recogerme” should also work.

Yes, it works. But there is a difference: ‘recoger’ usually means that you go to pick up a person in a car, on a motorbike, etc. and then leave together. ‘Buscar’ does not.
I don’t know if I have explained myself well, if not, let me know and I will try to explain myself better.

I understand. So, strictly speaking, “recogerme” is rather “pick me up”.

Yes. Recogerme=pick me up.