The Colombian cost Madrid more than £60m following his impressive World Cup, and the former Monaco attacking midfielder has assured the club's supporters that he is determined to enjoy a long and successful career in the Spanish capital.
"I always wanted to be here, I dreamed of it a lot. From the age of three or four I dreamed of being here and I want to stay here for many years," Rodriguez told Spanish radio station Cadena SER.
"I want to fight to the death so that this dream continues. I knew that I had talent, but in football you also have to have luck."
Rodriguez helped Madrid earn a 4-1 victory over Cornella in the Copa del Rey on Wednesday night.