I'm posting this to put Rahul's answer, given in a comment to the question, on record.
ContourPlot[Im[x + I y - Log[x + I y]], {x, -6, 8}, {y, -6, 6},
Contours -> {0},
RegionFunction -> Function[{x, y}, Re[x + I y - Log[x + I y]] > 0]]

The yellow regions are positive and the blue regions are negative.
The plot can be much cleaner if a nicer ColorFunction
is used and the cutoff at Re[function]==0
is not so abrupt:
ContourPlot[Im[x + I y - Log[x + I y]], {x, -2, 6}, {y, -6, 6},
Contours -> 16, (* 16 contours seemed like a good number *)
BoundaryStyle -> {Thick, Gray}, (* this adds a nice line at the edge of the region *)
RegionFunction -> Function[{x, y}, Re[x + I y - Log[x + I y]] > 0],
ColorFunction -> "TemperatureMap" (* cool colors are negative, warm - positive *)
]

There's still an artifact at y==0 && x<0
. This is to be expected, Log[z]
has a branch-cut on the negative real half-line.
ContourPlot[Im[x + I y - Log[x + I y]], {x, -6, 8}, {y, -6, 6}, RegionFunction -> Function[{x, y}, Re[x + I y - Log[x + I y]] > 0]]
$\endgroup$ – user484 Jan 13 '16 at 21:58