I'm never sure how to do these, minus the tedious algebra. I'm trying to see if I can develop a healthy intuition concerning problems of the kind; I'll summarize my approach here.
Let $w = 2z, z \in \mathbb{C}.$
Then, $0 < |2z - 1| < 2 \iff 0 < |w - 1| < 2$. The latter describes the interior of a circle of radius $2$ centered at $1 + 0i$ and excluding that point itself.
Making the appropriate adjustments for our substitution, we see that $\Re (z) \in (-.5, 1.5), \Im(z) \in (-\sqrt3/2, \sqrt3/2)$, and the center is given by $2z - 1 = 0 \iff z = .5 + 0i$. This describes the ellipse of minor radius $\sqrt3/2$, and major radius $1$, with the center $.5 + 0i$.
Is this correct? Also, would you recommend a better way of doing this?
Thanks to all in advance.