When discussing Quantum physics, the way it is described is almost always in terms of the Copenhagen Interpretation derived from the work of Neils Bohr, among others. There is another interpretation, the many-worlds interpretation (MWI), that was first suggested by Hugh Everett in 1957, and is essentially a reinterpretation of Niels Bohr's theories. The MWI assumes that the wavefunction of any object complies with the following:
This is totally at odds with the Copenhagen interpretation of quantum theory. In fact, at John Wheeler's suggestion, Everett went to Copenhagen and met with Niels Bohr, the "father" of the Copenhagen interpretation. It was, apparently, a disaster with neither man able to comprehend the other's stance. One colleague of Bohr's described Everett as an idiot, which was a little unreasonable!
In practice, this means that whenever there is an interaction, the wavefunction of the Universe branches into several paths; one for each possible outcome. This could be a macro event, I buy the shoes/I do not buy them, or an event at the quantum level, a particle decays/does not decay. As an example, consider the famous thought experiment of Schrödinger's cat. The Copenhagen interpretation tells us the cat is in a state of being neither-alive-nor-dead until an observer looks in the box and collapses the wave function. Many-worlds tells us that the universe splits into two; one where the cat is alive, the other where it is dead. In the first, the observer finds a live cat, in the second it is, unfortunately, a dead cat. Every possible outcome of every event exists within its own world, resulting in a virtually infinite number of universes, representing every possible thing that could have happened since the Universe began.
Today, many scientists are re-visiting Everett's ideas. Here is a brief biography of Everett, and discussion of the many worlds interpretation.
A rather more thorough discussion of the MWI is available here at Stanford.
This is another analysis by someone who is strongly supportive of the MWI.
||It is not observer-dependent,
It obeys the standard wave equations at all times whether a measurement is being taken or not, and
that the wavefunction is a real object in itself. That is, the observer has no special role and that there is no collapse of the wavefunction.