This is not a new idea. Connecting countries, and indeed the world, by underwater tunnels has fascinated science fiction writers and engineers for centuries. Jules Verne wrote about underwater travel and underwater pneumatic tube systems can be found in the world of Futurama.
Less fictional, there is of course the Channel Tunnel, now star of a new TV series, The Tunnel. And more recently opened, is the much awaited and controversial Bosphorus Tunnel, connecting Asia and Europe. Crossing the Atlantic though has proved more difficult. In the Daily Mail report on Gray's fantastical global network, engineer Robert Benaim suggests that one solution to the perplexing puzzle of how to connect America and Europe, may be a floating pneumatic tube, similar to Elon Musk's Hyperloop. This sounds very similar to Futurama's underwater system, and those of Verne's novels. Once again, fiction and real-life blurs, in the wonderful world of pneumatic tubes.
Image 1 my own, under the Bosphorus one month after tunnel opens in Istanbul, December 2013
Image 2 from Futurama