Finding conditions for which $z^{w_1}z^{w_2} = z^{w_1+w_2}$ is satisfied, and how strong are these conditions? I am trying to prove the claim for three complex numbers $z, w_1, w_2$ that  $$z^{w_1}z^{w_2} = z^{w_1+w_2} \iff \text{for any } k_1,k_2 \in \mathbb{R} \text{ there exists } m,k \in \mathbb{R} \text{ such that }$$
$$w_1(k_1 - m) + w_2(k_2 - m) = k$$
Firstly I rewrite each complex number in the form $z^\alpha = \exp[\alpha log(z)]$ and expand the $log(z)$ term:
$$z^{w_1} = \exp[w_1(\ln|z| + iArg(z) + 2k_1\pi i)]$$
$$z^{w_2} = \exp[w_2(\ln|z| + iArg(z) + 2k_2\pi i)]$$
$$z^{w_1 + w_2} = \exp[(w_1 + w_2)(\ln|z| + iArg(z) + 2m\pi i)]$$
Now if I multiply the first two together, I get 
$$z^{w_1}z^{w_2} = \exp[(w_1+w_2)(\ln|z| + iArg(z)) + w_12k_1\pi i + w_22k_2\pi i]$$
For $z^{w_1}z^{w_2}$ and $z^{w_1+w_2}$ to be equivalent, I must have $$w_12k_1\pi i + w_22k_2\pi i = (w_1 + w_2)2m\pi i + 2k\pi i \text{ for some } k \in \mathbb{Z}$$
Simplifying this, I get
$$w_1k_1 + w_2k_2 = (w_1 + w_2)m + k$$
Which can be arranged to $w_1(k_1 - m) + w_2(k_2 - m) = k$.
Is my approach correct? I'm quite confused about whether my proof meets the condition for any $k_1$ and $k_2$ there exists $m$, since I define $k_1$, $k_2$ and $m$ in the same way. 
And in general, how "strong" is this condition? I'm wondering how common it is for complex numbers $w_1, w_2$ to have this property where $w_1, w_2 \notin \mathbb{Q}$.

Looks ok. I am adding this here because comment space is not enough.
$k_1-m$ and $k_2-m$ are both $\in\mathbb{Z}$, so your last equation is essentially equivalent to the constraint:
$$k_1\cdot w_1+k_2\cdot w_2=k,\,\,\,\,k_1,k_2,k\in\mathbb{Z}$$
Susbtituting now $w_1=x_1+y_1i$, $w_2=x_2+y_2i$, into the above, gives you the system:
$$
        \begin{pmatrix}
        k_1x_1+k_2x_2 = k \\
        k_1y_1+k_2y_2 = 0 \\
        \end{pmatrix}
$$
Solving (for $k_1\neq 0$) gives:
$$
        \begin{pmatrix}
        x_1=\frac{-k_2 x_2+k}{k_1}\\
        y_1= \frac{-k_2 y_2}{k1}\\
        \end{pmatrix}
$$
and this gives you the wanted constraint in terms of the pairs $(x_1,x_2)$ and $(y_1,y_2)$ and hence in terms of the pairs $(w_1,w_2)$.
Ergo, the complexes $w_1$ and $w_2$ lie on two intersecting lines, with the intersection at the origin when $k=0$ (in which case $w_1$ and $w_2$ are conjugates).
Some Maple code to see what's going on:
restart;
with(plots);
w1 := x1+I*y1;
w2 := x2+I*y2;
k1 := 2; k2 := 7; k := 1;#change for different output
eq1 := k1*x1+k2*x2 = k;
eq2 := k1*y1+k2*y2 = 0;
solve({eq1, eq2}, {x1, y1});
#parametrize
X := proc (t) options operator, arrow; (-k2*t+k)/k1 end proc
Y := proc (t) options operator, arrow; k2*t/k1 end proc
plot({X(t), Y(t)}, x = -4 .. 4, view = -4 .. 4);

The plots below are for $(k_1,k_2,k)$:
1,0,0:

1,1,0:

1,1,1:

1,2,0:

1,2,3:

1,-2,3:

