Solution:
Let $z=re^{i\theta}=r(\cos \theta + i\sin \theta)$. Then $z^{1+i}=e^{i\theta(1+i)}=e^{i\theta -\theta}=4$. So $e^{i\theta-\theta}=e^{-\theta}e^{i\theta}=e^{-\theta}(\cos \theta + i\sin \theta)=4(\cos 0 +i\sin 0) \Longleftrightarrow e^{-\theta}=4 \quad \text{and}\quad \theta =0$. But, $e^0 =1$. What am I doing wrong?
EDIT 10/26/18
Would like to provide a more systematic and traditional approach:
\begin{align}
r^{1+i}e^{i\theta(1+i)}&=r^{1+i}e^{-\theta}(\cos\theta + i\sin\theta)=4(\cos 0 +i\sin 0)=4\\
r^{1+i}e^{-\theta}&=4 \quad \text{and}\quad \theta=0\\
\log_e(r^{1+i}e^{-\theta})&=\log_{e}(4)\\
\log_(r^{1+i})+\log(e^{-\theta})&=\log_{e}(4)\\
(1+i)\log_{e}(r)-\theta&=\log_{e}(4)\\
\log_{e}(r)&=\frac{\log_e(4)}{1+i}\\
\log_{e}(r)&=\log_{e}(4^{\frac{1}{1+i}})\\
r&=4^{\frac{1}{1+i}}\\
r&=4^{\frac{1-i}{2}}=[4^\frac{1}{2}]^{1-i}=2^{1-i}
\end{align}
Best Answer
$$4 = 2^2 = z^{1 + i}$$
$$2 \log 2 = (1+i) \log z$$
$${2 \over 1+i} \log 2 = \log z$$
$$2^{2 \over 1 + i} = z$$
$$2^{2(1-i) \over (1+i)(1-i)}$$
$$2^{2(1-i) \over 2} = z$$
$$2^{1 - i} = z$$
Check:
$$\left( 2^{1-i}\right)^{1+i} = 2^{1+1} = 2^2 = 4$$