Me tienes a mí

Me tienes a mí
 

Oí "me tienes a mí", pero no lo entiendo. ¿Por que no es "me tienes"? Parece que esta usando un objeto indirecto, pero normalmente "tener" usa objeto directo, ¿verdad?

I think it is another case of "reduplicación obligatoria" (mandatory redundancy): when you use a pronombre tónico you have to use also a pronombre átono.

Di el mensaje a mis vecinos ---->

Di el mensaje a ellos :bad:
Le di el mensaje a ellos :good:

Entregó las pruebas al juez ---->
Entregó las pruebas a él :bad:
Le entregó las pruebas a él :good:

Recuerda que me tienes a mí; recuerda que cuentas conmigo.:good:

You may find also examples without the reduplicated pronoun, but there's some focus in the beginning of the sentence (reduplication is accepted too): "No a muchos (le) han podido deber tanto dinero como a él"

By the other hand "Recuerda que me tienes" is ambiguous. What is it? "me tienes cansado"? you own me? you have chained me to a pole? "Tener + participio" is a verbal periphrasis so the sentence looks kind of incomplete. Also the "got me" part is not clear, is it my body? my soul? my will? That "me tienes a mí" implies my affectioned support.

The a mí was added for emphasis. And that's the only way to restate the direct object pronoun me. The preposition a is used because me references a person. This is also called the personal a. Since this is a preposition, an object must follow. When a personal pronoun is used as a prepositional object, it often takes a different form than when it is used as a direct object.

Have a look at the table of Spanish pronouns on this page for more information.