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:
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.