distances - How close would quasars have been to each other?

That's an image of two quasars, very close to each other, "separated on the sky by only about 70 thousand light years".

According to this paper the peak density should have been about $10^{-6}$ quasars per Mpc, resulting in a peak mean distance between two quasars of about 100 Mpc. (Density of $10^{-6}$ quasars per Mpc results in one quasar per cube of $10^6 Mpc^3$, hence on average one quasar each $sqrt[3]{10^6 mbox{Mpc}^3}=100 mbox{ Mpc}$)

With about 1 million known quasars in the observable universe I get a mean distance of about half that distance. (With a considered light-travel distance of about $4 mbox{ Gpc} = 4cdot 3.26 mbox{ Gly} = 13.04mbox{ Gly},~~$ a little less than the 13.81 Gly light travel radius of the observable universe, the volume of a sphere with this radius is about $frac{4}{3}pi(4cdot 10^3 mbox{Mpc})^3=2.68cdot 10^{11}mbox{Mpc}^3$. With 1 million quasars, it's about one quasar per $268000 mbox{ Mpc}^3$ or one quasar per $sqrt[3]{268000 mbox{ Mpc}^3}=64mbox{ Mpc}$ in one dimension. Due to non-constant quasar frequency over cosmic time, the minimum mean distance should have been below this overall average distance. Using the light-travel distance considers the higher density of the earlier universe better than the comoving distance of 13.8 billion years after big bang. 4 Gpc are used as light travel radius to restrict roughly to redshift z < 7, the range where quasars have been observed. A redshift calculator can be found e.g. here, take $H_0=70.4$, $Omega_m = 0.265$, $Omega_{Lambda}=0.728$, and $z=7$.)

The peak number density seems to be somewhere between a redshift of 2.0 and 2.5 (interpreting the paper). If it will turn out by future observations, that the peak number density has been earlier in the universe, we get much shorter peak mean distances.

So you have at least an order of magnitude: Somewhere between 150 and 300 million lightyears (between 50 and 100 Mpc, 1 parsec equals 3.26 lightyears) should have been the peak average distance between two quasars. Might be this estimate will shrink over time, as more (distant) quasars get known.