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> We have explicit expressions for the quadratic Hilbert symbol over $\mathbb Q$, for example $\left(\dfrac{x,y}2\right)\_2=(-1)^{\frac{x-1}2\frac{y-1}2} (x,y\ne2)$. Are similar expressions known for the Hilbert symbol $\left(\dfrac{x,y}v\right)\_4$ of order 4, for the places $v$ of $\mathbb Q(\sqrt{-1})$? In particular for the place above 2?
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For completeness I recall the definiton:
Using the notation from Gras' *Class Field Theory*, the Hilbert symbol $\left(\dfrac{-,-}v\right):K^\times\times K^\times\to\mu(K)$ on a global field $K$ at the place $v$ is given by
$$\left(\dfrac{x,y}v\right)=\dfrac{\sigma(\sqrt[m]{x})}{\sqrt[m]x},$$
where $\sigma=\left(\dfrac{y,K(\sqrt[m]x)/K}v\right)$ $(m=\#\mu(K))$ is the Hasse symbol, i.e., the image in $Gal(K(\sqrt[m]x)/K)$ of the local reciprocity map applied to $y$ for the given extension. For $n$ a divisor of $m$, the Hilbert symbol of order $n$ is 
$$\left(\dfrac{-,-}v\right)\_n=\left(\dfrac{-,-}v\right)^{m/n}.$$



Computation of Hilbert symbol of order 4
Yes, such formulae are known.
The general case was first stated by Brückner, a somewhat more modern statement was given by Henniart: <http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN243919689_0329&DMDID=DMDLOG_0017&LOGID=LOG_0017&PHYSID=PHYS_0183>


A warning, though: The formulae in the ramified case, especially when $p=2$, are horridly complicated


