I wonder if you can create such an algebra using matrices. The determinant of matrices can positive or negative.
Let 1 = ((1,0),(0,1)) [the 2x2 unit matrix] and c = ((0,1),(1,0)). Define a crazy number as z = a * 1 + b * c, where a is the real part, and b is the crazy part. Now identify an ordinary number x with Sqrt[x] * 1. And define the absolute value of a crazy number z as Det[z].
Now for any real number x we have |x| as a crazxy number is |x| as a real number. And |c| = -1. That's what we wanted isn't it?
This does mean that |1 + c| = 0. That's rather strange
edit: I forgot to mention. Many properties of absolute values remain intact using the above definitions. Most importantly |z1 * z2| = |z1| * |z2|.