It all started with a math-geek desire to come up with an "imaginary" dollar-- a unit of currency equal to the square root of debt. Unfortunately, googling "imaginary dollar" just gets a load of people saying that money is already imaginary. :) So, I eventually googled "hypercomplex dollar" (the hypercomplex numbers go beyond imaginary numbers, introducing things like hyperbolic multiplication, et cetera). I got this:
Sadly, this and the other proposals it links to are not really attempts to construct imaginary money: they just use the fact that the imaginary numbers (and hypercomplex numbers) are orthogonal to real money; they don't attempt to give an answer to what the square root of a monetary amount might be.
Anyway, these proposals referenced the LETS money system, which is interesting...
The basic idea: everyone starts out with 0 money, but is free to spend anyway. This puts them in the negative, but negative is referred to as "commitment" rather than debt, and is not supposed to be a bad thing. There is always as much negative as positive, so being in the negative is normal and does not cause a problem (since you're still free to spend even more). The justification: scarcity of money causes a lot of problems. Scarcity of dollars stops local economies, even if there is really enough goods and labor to go around. This is why more US dollars are printed then go out of service each year: to fight the scarcity of money. Unfortunately, that leads to inflation, which has its own problems... LETS tries to fix this by letting anyone issue money whenever they like, creating "commitment" in doing so. This means there doesn't have to be money around in order for people to exchange commitment for goods and labor, so that money scarcity can never be a problem.
This basically relies on individuals to honor their commitment. I think this is a big part of why LETS is supposed to remain a local currency: it relies on the goodness of people, so if it got too big it would run into troublemakers.
My take on the future of money is that we are headed for a credit-based economy (in which credit is more important than raw money). We're already almost there! However, there are some serious flaws with the current credit economy... credit cards are a Bad Plan. Personally, I think things like kickstarter are a better sign of what's coming... people investing in people. The problem with banks and credit cards is that all the investing is being done by a few monolithic entities, whereas the "little guy" just gets the debt. The little guy needs to be able to give credit as readily as take it.
The idea of "debt" is owing money to one person. The LETS idea of "commitment" is to instead owe to a community. The problem is that this is too easy to take advantage of it becomes too mainstream (ie, if people don't feel a personal commitment to the whole community). The currency would just inflate.
To address this, I propose an idea of financial backers. Individuals back each other rather than being backed by a big bank. If a group of people back each other with no reservation, then they should automatically form something like a LETS network: they can spend as many credits with each other as they want, making "commitment" with the group as they do so. When they get money, either within the group or from outside, the commitment gets payed off.
More commonly, people could offer each other limited lines of credit. The amount I offer to you would depend on a combination of how much I trust you, how much I can afford personally, and how much credit I have from other sources.
Obviously more details need to be worked out here. One principle from LETS is that no interest should be charged on commitments. It's not obvious whether that's preferable in the system I'm thinking about. It promotes a positive and friendly outlook (as does the option of offering an unlimited credit line to people you trust, rather than worrying about placing a $ cap). However, it also seems to make sense to pay people back for their hospitality... perhaps mutual credit agreements are better than interest for that purpose, though.
One conspicuous advantage of this scheme (in contrast to the currently mainstream big-bank approach!) is that the distributed nature of the system prevents large failures. An uncareful collective of individuals might fall together, but the impact would be limited.
EDIT: This proposal has already been thought of and is being implemented! If you extend me a little credit in ripplePay I'll extend you a little back. :) I may write a post soon on the differences between what I was imagining and what these people are doing.