Ttgjyuk

With the following code:

documentclass[tikz]standalone

usepackagetkz-euclide,tkz-fct,amsmath
usetkzobjall


begindocument
 begintikzpicture[anchor=center]
 tkzInit[xmin=-1, xmax=3, ymin=-1,ymax=3]
 tkzDefPoints.5/2/P_1, 2.5/0/P_2, 1.5/1/M,2/1.5/A

 tkzDrawX[noticks, label=(operatornameRe(x) )]
 tkzDrawY[noticks, label=(operatornameIm(x) )]

 tkzDrawPoints[fill=black, size=1mm](P_1,P_2,M)
 tkzMarkRightAngle(A,M,P_1)
 tkzFct[domain=-1:3, color=red, thick]x-.5
 draw (P_1) -- (P_2);
 tkzLabelPoints[above right](P_1,P_2)
 tkzLabelPoints[right](M)

 tkzText[color=black](1.5,3)z-z_1
 endtikzpicture
enddocument

I'm getting:

I wanted to add a fade like this:

but I can't have the fade to be in the right angle.

How can I get this kind of fade, fadding to white?

You may rotate the shading area to the x-axis, shade, then rotate back with 'transform canvas'

documentclass[tikz,border=5mm]standalone
usepackagetkz-euclide
usetkzobjall
begindocument
begintikzpicture
coordinate[label=above right:$P_1$] (P1) at (.5,2);
coordinate[label=above right:$P_2$] (P2) at (2.5,0);
coordinate[label=right:$M$] (M) at (1.5,1);
coordinate (A) at (2,1.5);
pgfmathsetmacroa.5-sqrt(2)
pgfmathsetmacrob.5+sqrt(12.5)

shade[top color=white,bottom color=red!50,
transform canvas=rotate around=45:(.5,0)] 
(a,0) rectangle (b,.5);
draw[-latex] (-1,0)--(3.5,0) node[below]rmRe$(x)$;
draw[-latex] (0,-1)--(0,3.5) node[left]rmIm$(x)$;

draw (P1) -- (P2);
draw[red,thick] plot[domain=-.5:3] (x,x-.5);
foreach p in P1,P2,M
fill (p) circle(1pt);
node at (1.5,3)z-z_1;
tkzMarkRightAngle(P1,M,A)
endtikzpicture
enddocument

This is in principle very simple but tkz-euclide seems to mess up things a bit. One can just use a shading angle, which can, of course, be computed by TikZ.

documentclass[tikz]standalone
usetikzlibrarycalc,backgrounds
usepackageamsmath
DeclareMathOperatorreRe
DeclareMathOperatorimIm
begindocument
 begintikzpicture[anchor=center,declare function=f(x)=x-0.5;
 xmin=-1;xmax=3;]
 draw[-latex] (-1.5,0) -- (3.5,0) node[below left]$re z$;
 draw[-latex] (0,-1.5) -- (0,3.5) node[below left]$im z$;;
 path foreach X/Y/L/P in .5/2/P_1/45, 2.5/0/P_2/45, 1.5/1/M/0
 (X,Y) coordinate[label=P:$L$] (L);
 beginscope[on background layer]
 shade let p1=(xmin,f(xmin)),p2=(xmax,f(xmax)),
 n1=atan2(y2-y1,x2-x1) in 
 [left color=white,right color=red,middle color=white,shading angle=n1]
 (p1) -- (p2) -- ($(p2)!2cm!-90:(p1)$) -- ($(p1)!2cm!90:(p2)$)
 ;
 endscope 
 draw[red,thick] plot[variable=x,domain=xmin:xmax] (x,f(x));
 draw (P_1) -- (P_2);
 node[anchor=south,red] at (1.5,3) $;
 endtikzpicture
enddocument