Neural chat translation (NCT) aims to translate utterances in a bilingual dialogue, where participants speak different languages, into their respective native languages. Recent advances on NCT translate chat utterances one by one using document-to-sentence (Doc2Sent) models, associated with deliberately designed auxiliary tasks that explicitly model the inherent bilingual conversational characteristics. Different from Doc2Sent models, in this paper we simply view chat translation as online document-to-document (Doc2Doc) translation, in which both the input and output are bilingual. Moreover, in order to alleviate the challenges that the encoder and decoder may encounter in grasping semantics with bilingual context, we propose a cross-lingual regularization which regularizes the translation prediction between the default bilingual-input translation and two auxiliary monolingual-input translations. Experimental results on English \(\leftrightarrow \) German and English \(\leftrightarrow \) Chinese chat translation tasks show that based on G-Transformer, a state-of-the-art Doc2Doc translation model, our approach outperforms sentence-level baselines and many existing models.