While you could argue rather or not it constitutes a 'new' government, or just an expansion of the existing one, I have asked two previous questions about how to create a hybrid of direct and Representative democracies which feels like a very new take or the existing democracy, perhaps counting as it's own form.
The linked question addresses some of the concept, but let me go into detail. The idea is that with the internet we are capable of near-instantaneous voting now, making a direct democracy more fiesable then it has ever been before. Of course currently voter fraud and hackers are an issue with online voting, but we could presume in the future these are addressed and a system can be created which is no more prone to such fraud then modern voting, and that internet access has grown to be such a staple that it's presumed to be available to even the poorest citizen. If this is the case we can start to craft a system which allows voters to more directly express their wishes by voting directly on important issues.
The advantage to a more direct vote is that votes can be more representative of an individuals opinion, where as it's impossible to pick a perfect representative that truly represents your views. Maybe you mostly agree with political party X, but your views on abortion are opposite of the party line and you have a slightly different view on the appropriate level of gun regulation. You pick the senator from your political party because you agree with 85% of that parties views, but that means that you picked someone who is going to do his best to make abortion laws that are the exact opposite of your moral views. You picked the best representative you could, and still can be upset that your views on abortion are not being reflected. With a more direct democracy you can vote along the party line for the 85% you like, and vote against them when it comes to abortion, better representing your own personal beliefs.
f course the problem with this is that voters don't have the time to vote directly on every issue. a pure direct democracy would be ruled by those that took the time to vote (like the elderly) and would thus leave a huge percentage of people unrepresented. Already most only vote for president, even picking a senator is rare, but that is only one decision every few years. Imagine if you were asked to make a decision every day, few would do it, making those who did quite powerful.
Thus you create a hybrid solution, with representatives picked by voters who actually manage most votes for a voter, but with the voter able to personally vote on any law or legislation he chooses as well.
My question went further to suggest the idea of voter blocks and a sort of hash-tag approach to voting. Each voter can pick a representative to represent them for certain types of votes. For instance you may have someone who believes in small government with little overhead (like republicans) but in more liberal 'if it isn't hurting me let it be' social policies, like the democrats. He may pick a republican 'fiscal' representative and a democratic 'social' representative, who will vote according to his views. Social matters thus get one vote and fiscal a separate vote.
In reality things would likely grow more complex, with one voter able to grant his voting rights out to many representatives. To make this work imagine a system where each bill is given a number of, well frankly government standardized hash tags, to represent the content of the bill. The bill to increase taxes and move the money into supporting faith-based charities may be tagged as fiscal, but with a sub-tab of religion and 'government-support' (some tag similar to welfare, but with a more generalized meaning).
A voter can assign a representative to represent a tag he cares about, but he gets more flexibility then that. This system requires a vote to be divisible, one vote may be infinitely divided. So our theoretical voter who preferred small-government may have chosen the small-government fiscal representative. However, he may also be very devout and so for the religious sub-tag he picked a strong "christian rights" representative.
For this vote his fiscal voter may vote no for the bill, taxes mean big government, but his religious representative may vote yes, more money to faith based initiatives help them to do god-work. One vote each way effectively nullifies his vote, and so his vote doesn't work at all.
However, he may have more flexibility in how he assigns his votes. Maybe he chose the fiscal rep as his main voter, but set it up so that in times where it conflicts the religious rep gets priority by representing 2/3 of his vote on religious bills. The fiscal rep votes against the bill, with 1/3 of the voters granted vote, and the religious rep votes for it with 2/3 of the voters proxy vote, leading to that voter voting 'yes' with 1/3 of a vote. Modern vote systems can easily handle such fractional votes, enough 1/3 votes can counteract full votes...
Of course perhaps in this situation our voter feels that it's a bad idea. He figures that the letter of the law means that christian groups could not use the extra funding from the government to actual preach or evangelism, so the law doesn't help him spread his religion and, to him, is just another form of welfare. So he decides to vote directly against the bill, taking back his proxy vote and directly voting against the bill. He can, at any point, do this, his representatives are simply a way of voting by proxy for the votes he can't be bothered to vote on directly; he always has the right to directly vote personally on any bill he considers important enough to vote on.
That's the short idea. This will result in a very different feel, with a decrease in power of political parties as instead of voting on specific large platforms voters can focus on personal belief systems. However, complex politics will start to be about how to assign the 'tags' to a particular bill in order to get the 'right' representatives to vote for the bill, ie the ones you think will vote in your favor.
For example, say there is a a vote being made about rather to fund abstinence based sex-education or to fund only comprehensive sex-education. Perhaps those that want to pass the comprehensive sex-education bill will try to get the "LGBT-rights" tag assigned to the bill, on the grounds that comprehensive sex-education is more likely to address alternate sexuality and therefore pro-LGBT representatives would therefore be more likely to vote for the bill. You could argue rather sex-ed really is an LGBT-rights issue, but rather or not it gets assigned to a bill is partially based off of rather one group or another thinks those reps will vote in their favor. Sadly you can't get the politicking out of politics lol.
Notice I've refereed to finance votes in a way that isn't really representative of how bills work currently in the US, that's intentional. In this new system there would likely be a focus on breaking up larger decisions, like funding decisions, into a number of smaller votes so that people can better vote on them individually; which would also require a focus on less time debating each bill if you want anything to be done, or perhaps just more parallel votes going on in different sectors, the 'financial' and 'social' votes can happen mostly independently since reps for one can vote separately from reps of the other.
Of course this addresses only the votes. Someone has to craft the original bills and votes. Thus I imagine a representative will be picked the way most representative democracies do who is responsible for crafting and modifying bills, and this rep's vote would count for anyone who has not transferred his voting proxy to another representative. The difference is that when votes happen multiple voting representatives can vote along more complex divides then just bi-party views.
I imagine one of the biggest bit of politicking in this system would be how to assign 'tags' to a given vote. A group may try to get certain tags attached to a vote because they know that the majority of voters for that tag will vote along the lines they want, so bickering over what tag to assign or not assign may become quite common. This is assuming that the government assigns tags. As an alternative there may be non-political entities that read and assign tags and someone may choose to use any one of a few 'tag assigning' entities as the one that decides how to tag a bill to decide how their vote proxy should be divided.
Obviously this system only works in a modern world with computers. This would all get way to complex without not just instantaneous voting, but also the ability to help calculate complex rules for how a single vote was divide. However, with modern computers these sort of calculations can be done easily. All a user has to do is assign his vote proxy using what would amount to a simple rules-engine to best define how he wants to divide up his vote amongst representatives. Most will assign their vote to a single rep, some will assign their vote to a single rep but with one or two issues that are important to them getting special reps, and only a few will right more complex rules for vote proxying, but every voter gets to decide how they wish to divide up their vote, and every voter gets the right to personally vote on an important vote if they choose to.
*note, these theoretical voters do not necessarily represent my personal political views, I'm personally a registered independent without strong preference to either party. I added views as I came up with a good example that would require someone having a certain view that's all.