Step 1: Use that $z\bar z$ is real.
For any complex number, $z\bar z = |z|^2$ is a real number. We will use this to pin down the form of $z$.
Step 2: Clear the fraction.
We are given $\bar z = \frac{1}{z - i}$. Multiply both sides by $(z-i)$: \[ \bar z (z - i) = 1 \implies |z|^2 - i\bar z = 1 \]
Step 3: Isolate the $i\bar z$ term.
Rearrange: \[ i\bar z = |z|^2 - 1 \] The right side is real, so the left side $i\bar z$ must be real too.
Step 4: Force $z$ to be imaginary.
Write $z = x + iy$, so $i\bar z = i(x - iy) = y + ix$. For this to be real we need $x = 0$. So $z = iy$.
Step 5: Plug back in.
With $z = iy$, the relation $\bar z(z-i)=1$ gives $-iy\,(iy - i) = 1$, which simplifies to \[ y(y-1) = 1 \implies y^2 - y - 1 = 0 \]
Step 6: Solve for $y$.
By the quadratic formula $y = \frac{1 \pm \sqrt5}{2}$. So one allowed value is \[ \boxed{ z = i\left(\frac{1+\sqrt5}{2}\right) } \]