How to sketch the subset of a complex plane? The question asks to sketch the subset of $\{z\  \epsilon\  C : |Z-1|+|Z+1|=4\}$  
Here is my working:
$z=x+yi$
$|x+yi-1| + |x+yi+1|=4$
$\sqrt{ {(x-1)}^2 + y^2} + \sqrt{{(x+1)}^2+y^2}=4$
${ {(x-1)}^2 + y^2} + {{(x+1)}^2+y^2}=16$
$x^2 - 2x+1+y^2+x^2+2x+1+y^2=16$
$2x^2+2y^2+2=16$
$x^2+y^2=7$
$(x-0)^2+(y-0)^2=\sqrt7$
=This is a circle with center $0$ and radius $\sqrt7$  
My answer is different from the correct answer given: "This is an ellipse with foci at $-1$ and $1$ passing through $2$"
I have no idea how to get to this answer. Could someone please help me here? 

That's a good question. Unfortunately you can't just square term by term like that. When you write this out, let z=x+iy. If you do that you will get a very complicated algebraic term. 
((x-1)^2 + 2*y^2 + (x+1)^2)^2 = 16
As LutzL had brilliantly stated you can make a substitution, in this case a make a parametrization, where a = (x-1)^2 and b = (x+1)^2
However there is still a much easier solution. If you have a textbook called Advanced Mathematics by Terry Lee, he goes through how this can be done. This is an ellipse. The 4 at the end of the equation, instead of representing the radius of a circle, this represents the major axis of an ellipse.
We know for an ellipse x = 2a (this is equation of major axis).
Hence 2a = 4              Therefore a = 2
We now know our semi-major axis is 2. If we use form of an ellipse where 
(x^2)/(a^2) + (y^2)/(b^2) = 1 
Hence we can write (x^2)/4 + (y^2)/(b^2) = 1
The foci represents the left part of the equation in this case going from z-1 to z+1. Set these to equal 0 and hence the foci are (1,0) and (-1,0).
Now let S = focus
The equation of the focus is given by: S = ae
We substitute our data: S=1, a=2(semi-major axis)
e=1/2 (eccentricity)
From this we can then find the semi-minor axis.
We know the formula b^2 = a^2(1-e^2)
If we solve, we get b= sqrt(3)
Hence the equation of the ellipse is given by: x^2/4 + y^2/3 = 1
If you want you can now find directrices: Let D = Directrix
By definition D = +- a/e = +- 2/0.5 = +- 4
You can now graph this.
