Find minimum value in complex plane? Minimum value of
$$|z+1|+|z-1|+|z-i|$$ for all $z$ from complex numbers? 
Is there any particular way to handle these type of problems?

There may be more efficient ways, but here's an approach that doesn't require knowledge of the Fermat point in the comment.
Substituting $z=a+bi$ and using $|x+yi| = \sqrt{x^2+y^2}$ turns this into a two-variable optimization problem in (the real numbers) $a$ and $b$. Rather nasty, however.
Since $|z-c|$ represents the distance from $z$ to $c$, your problem can be interpreted as minimizing the sum of the distances of $z$ to respectively the (complex) numbers $-1$, $1$ and $i$; make a simple sketch. For reasons of symmetry, it is clear that this minimum will lie on the (positive) imaginary axis, meaning we can take $a=0$ in $z=a+bi$ above, reducing the problem to a one-variable optimization problem in the (real) variable $b$ (with $0<b<1$).
$$f(z) \overset{z=a+bi}{\underset{a=0}\longrightarrow} f(b) = |bi+1|+|bi-1|+|bi-i|=2\sqrt{b^2+1}-b+1$$
Finding the minimum of
$$f(b) = 2\sqrt{b^2+1}-b+1$$
can be done with the usual technique (look for the zero of $f'(b)$) and yields $b=1/\sqrt{3}$ meaning that the initial expression attains its minimal value at $z=\tfrac{i}{\sqrt{3}}$.
