As already mentioned, currency won't be a sensible concept for the first wave of settlers, they'll just be trying to build an infrastructure. I'm not sure how much money gets used at Amundsen-Scott station, for instance.
Once there is a permanent human presence on Mars, and at least some of the inhabitants are producing extra food, fancy clothes, and works of art, they'll need a currency. It would be insane for that currency to depend somehow on how things are back on Earth, since there'd be almost no traffic of goods or people between the two. If the Martians did try to implement a Marscoin operating on the same principles as BitEarthcoin, they'd need their own internet, miners, genesis block, and subsequent blockchain. It would be fascinating to see how a variant of Bitcoin would develop in a 'virgin' environment, where it doesn't have to compete with existing currencies and all the dodgy things that go on with them.
But what would be the point of "splitting" the concept of bitcoin? As an organism mankind is moving forward. Although this is science fiction now, there will be a point in the future where we will look back at 2013 like the societies in Europe were before the discovery of the American continent.
The concept of a Marscoin is cool. But let's say something very rare and amazing exists on Mars. Then going there won't be a matter of being suicidal as you may never come back. Because of that new found rare Mars dirt, you may NEVER EVER want to go back on Earth. Then what would happen? The Marscoin would have a bigger value than the Earthcoin. Instead of Forex between countries people will bet against whole planets. transfers of technologies, minds, wealth will travel to Mars, leaving behind on Earth only those who can only afford to live with that very old Earthcoin, Marscoin being too valuable and too expensive. Maybe some could see their demise as being because of all those fat cats on Mars creating chaos on Earth They were the first ones mining the planet, risking their lives at first, and a lot did die, but now they are ruining the lives of those on planet earth. The one left behind.
Actually, if it was not for Earth, no one would have been able to even travel on Mars anyway. Some would demand a fair share of Mars fortune thanks to their ancestor's investments at risk. This could only escalate to its logical next step. War.
If the same protocol is used everywhere, no split will ever happen no matter where we ended up and those types of division will not repeat in the future like it did many times in our History's past.