Do "complex percentages" exist? Well, the origin of this question is a little bit strange. I dreamed - with a book called "Percentages and complex numbers. When I woke up, I thought: "Is this real?" So I started thinking:
1% of 100 = 1
3% of 100 = 3

And more:
i% of 100 = i?

That's my question. Is it right - does it even make any sense? Calculating a percentage is, basically, to multiply a fraction (denominator = 100) to a number. If it's right, we can, for example, calculate
8 + 4i% of 2 - i.

So, what you can tell me? Is it real or just a dream - have you ever seen this? And more: does it respect the definition of percentage?
Thank you.
 A: I tend to interpret % as shorthand for the fraction $\frac1{100}$; $i\%$ would just be $\frac{i}{100}$, and $(8+4i)\%$ would be $\frac2{25}+\frac{i}{25}$. As to whether these things are useful...
A: Whether this is "real" or "exists" is a philosophical, not a mathematical question. My personal opinion on such questions is that they are meaningless, that is, there is no way to distinguish a state of affairs in which this concept is "real" from one in which it is not "real".
Concerning your question "have you ever seen this?": No, I haven't, and I would hazard a guess that if anyone has ever thought of this, they haven't found a use for it yet. However, that doesn't keep you from doing it. The fun thing about mathematics is that you can define whatever you want, as long as it's consistent, and see what it leads to, without worrying about whether it's "real".
A: I am resurrecting an ancient question because I ran across it and thought of a random paper I had seen a few years ago.  The original conversation participants have long since moved on, so I'm posting this for the benefit of anyone making their way here from Google.
As @joriki points out, there is not necessarily anything "god-given" about complex percentages, but maybe someone might find a useful way of viewing such a beast.  (In fact, there may be multiple different definitions of complex percentages, where each might be useful in a different way.)  Here is a paper that discusses a possible interpretation of something related, which is negative probability:
http://www.wilmott.com/pdfs/100609_gjs.pdf
The paper discusses an interpretation of negative probabilities as corresponding to flipping "half a coin."  Specifically:

Fundamental theorem: For every generalized g.f. $f$ (of a signed
  probability distribution) there exist two p.d.f.’s $g$ and $h$ (of
  ordinary nonnegative probability distributions) such that the product
  $fg = h$.
Thus if $f$ is the generalized g.f. of a half coin C, a third of a die,
  (or any other related mystical object), then we can always find two
  ordinary coins, ordinary dice (ordinary random object) C1 , C2 such
  that if we flip C and C1 , their sum is C2 . In this sense every
  generalized (signed) distribution is a kind of difference (‘so-called
  convolution difference’) of two non-signed (ordinary) probability
  distributions. This result justifies the application of signed
  probabilities in the same sense as we use negative numbers.

So one can view a single coin flip as the succession of flipping another coin and a half-coin.  The probabilities associated with flipping the half coins are negative.  You're "not allowed" to stop after only flipping the a half coin, in the same way a negative balance in your bank account doesn't give you the ability to carry negative one $20 bill in your pocket.
The authors also briefly allude to the wavefunction in quantum mechanics, which might better correspond to your question.  One recovers a probability from the wavefunction by taking the square of the norm of the wavefunction.  So the wavefunction is a probability-related thing which may be complex.  At a (very) high level, the complex nature of the wavefunction allows you to do "bookkeeping" to simultaneously keep track of conjugate quantities such as position and momentum.
So the wavefunction might correspond to a kind of complex probability (complex percentage), although if you want to get anything physically measurable out of this "complex probability," you end up feeding it through a process that spits out a real number, without any imaginary part.  This is similar to the linked paper, where negative probabilities can be thought of as corresponding to half-coins, and the half-coin is a "bookkeeping" technique that doesn't make its way into your pocket next to single coins.
