When speaking of neighbourhoods in complex analysis, are we always referring to circular neighbourhoods? In Complex Analysis, does "neighbourhood" automatically mean "circular neighbourhood", or do non-circular ones exist?
 A: Certainly non-circular neighborhoods are allowed. Using Cauchy's Theorem it is often possible to replace an integral along a curve by an integral along a circle. But there is no need, and little sense, in limiting the discussion only to function defined on circular neighborhoods. Moreover, the concept of analytic continuation is extremely important and one often is interested in the maximal region a given function can be extended analytically to. The resulting region will often not be circular. 
A: It probably depends on the textbook you're using. Some authors write "neighbourhood" in the sense "open neighbourhood". Some do not.
Some authors might only consider open discs, but the most common convention is that an (open) neighboorhood of a point is any open set containing the point. In practice, you can almost always work with disc shaped ones though. (Every open neighbourhood of $z$ surely contains a disc centered at $z$.)
A: The term "neighborhood" of a point comes from topology, where it is taken to mean any open set containing that point. This convention carries over into complex analysis.
Often, a circular neighborhood of a point and a more general neighborhood of a point are interchangeable in practice. If you need a circular neighborhood of a point, it suffices to find a general open set containing the point, since then you can just take a circular neighborhood contained in that open set; and if you need a general open set, of course a circular neighborhood will suffice. So topologically there really is no distinction.
Besides, many theorems in complex analysis that use circular neighborhoods can be adapted somewhat to more general regions, the idea behind the Riemann mapping theorem.
