Let the complex number be $z = x + iy$ where $x,y \in \mathbb{R}$.

Then $z-i = x + i(y-1)$ so $|z-i| = \sqrt{x^2 + (y-1)^2}.$

Also, $z-1 = (x-1) + iy$ so $|z-1| = \sqrt{(x-1)^2 + y^2}.$

Given $|z-i| = |z-1|,$ therefore $\sqrt{x^2 + (y-1)^2} = \sqrt{(x-1)^2 + y^2}.$

Squaring both sides, $x^2 + (y-1)^2 = (x-1)^2 + y^2.$

Expanding, $x^2 + y^2 - 2y + 1 = x^2 - 2x + 1 + y^2.$

Cancel common terms $x^2$, $y^2$, and $1$: $-2y = -2x$ $x = y.$

The equation $x=y$ represents a straight line through the origin with slope $1$.

Hence the correct option is: $\boxed{\text{D: the line through the origin with slope }1}$

Comparison with stored correct answer: Stored correct answer is D, which matches the derived answer.