The German national and founder of the Megaupload file-sharing service, who has permanent residency in New Zealand, faces up to 20 years in jail if convicted in the United States of piracy, which US authorities say cost copyright owners hundreds of millions of dollars.
His lawyer Ron Mansfield told the court the case raised "unprecedented issues of public and international interest" and it would not be a fair hearing without live-streaming.
Streaming had been successfully used in previous court cases and inquiries in New Zealand, he added.
However, Judge Murray Gilbert delayed making an immediate decision, saying he wanted to give other media representatives a chance to consider Dotcom's request and make submissions.
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Dotcom, who arrived at the Auckland High Court after the hearing began, had previously tweeted that Washington "has asked the New Zealand High Court not to allow live-streaming of my global interest copyright hearing. Worried?"
It is nearly five years since the 41-year-old, a self-described "Internet freedom fighter", was arrested in a dramatic police raid on his mansion near Auckland in January 2012 after the FBI shut down Megaupload's servers, and the case could rumble on much longer.
In December last year, after a nine-week hearing, Judge Nevin Dawson found there was "overwhelming" evidence to support extradition of the 41-year-old and three other Megaupload founders.
A lawyer for two of Dotcom's co-accused -- Mathias Ortmann and Bram van der Kolk -- told the court Monday they were simply part of a group who "invented file storage".
He said the men have been falsely accused of wrongdoing by the United States government and unfairly subjected to "grossly-excessive search and seizure operations".