If you want a discrete colormap with nonuniformly-spaced breakpoints, you're going to have to make one. You're probably using a colormap with 256 breakpoints when you want 4. If your breakpoints were evenly spaced, you could do colormap(jet(4)), but since they are arbitrarily spaced, you're probably going to have to make a colormap such that breakpoints are where you expect them to be.
On the other hand, if you can extend caxis such that the breakpoints are evenly spaced, then you can just use a regular colormap.
x = (0:100)/100;
y = x';
z = (0.15+0.0094)*(x+y)/2 -0.0094;
contourf(x,y,z,[-0.05 0 0.05 0.1 0.15])
caxis([-0.05, 0.15]);
colormap(parula(4));
c=colorbar;
