As mentioned, the legal side of buying property in Argentina is straightforward. The only documents needed are your passport and your personal Argentine tax number - CDI (Clave de Identificacion).  To obtain a CDI you must prove your address in Argentina (rental apartment, friend’s house, etc…) with a “certificado de domicilio”. 

This can be obtained at the nearest police station to where you are staying. With this document and your passport you must then go to the nearest AFIP office (Argentine tax authority) to get the CDI. If you do not speak Spanish it is recommended that you go there with somebody who does speak the language and is also familiar with the system. Otherwise, you can let somebody else do it for you (requires a Power of Attorney).