I don't think you should see language as a dimension in the 'normal' sense of the word (dimension). However, i think you could see language as something that could be placed IN a dimensional scale of another sort: the mind can be seen as a complex of layers, using codes to proces all kind of things related to survival. Counsciousness uses codes, to refer to these codes. Then thoughts exist of codes that refer to the codes that refer to these codes. And then language again, uses codes that refer to the codes of thought, that refer to codes that refer to codes. And then there's  meta-language that uses codes to refer to language ( codes wich are language itself, usually distinguished by quotation marks).

You could say that language has a place within the dimensional scale of another universe, the universe of the mind.