Our conjecture is that multiplying a point on the Argand diagram by -1 is equivalent to a 180 degree rotation about the origin.

Let $(a,b)$ on the Argand diagram represent the complex number $a+\mathrm{ib}$, where $i$ is the square root of -1. So, multiplying $a+\mathrm{ib}$ by -1, we get $-a-\mathrm{ib}$, which is represented by the point $(-a,-b)$ in the Argand diagram. However, the point $(-a,-b)$ is a rotation of the point $(a,b)$ by 180 degrees about the origin. QED.

Using the same numbers for the second part, our conjecture here is that multiplying a complex number by $i$ gives a rotation of 90 degrees anti-clockwise on the Argand diagram.

Again, assuming $(a,b)$ is a complex number plotted on the Argand diagram, by multiplying $a+\mathrm{ib}$ by $i$, we get the complex number $-b+\mathrm{ai}$, which is represented by the point $(-b,a)$ on the Argand diagram. This is equivalent to a rotation of 90 degrees anti-clockwise about the origin. QED.

Assuming we were to multiply the number by $i$ twice, this would give a rotation of 180 degrees (as shown in the first part because $i^2=-1$). Multiplying by $i$ thrice would be equivalent to a rotation of 270 degrees anti-clockwise, or 90 degrees clockwise and four time maps the number onto itself (essentially multiplying by 1). Multiplying by $i$ $n$ times is equivalent to rotating the point 90 degrees anti-clockwise $n$ times. Therefore; if $n$ is 0 mod 4, it has no effect; if $n$ is 1 mod 4, it's equivalent to an anti-clockwise rotation of 90 degrees; if $n$ is 2 mod 4, it's equivalent to a rotation of 180 degrees; if $n$ is 3 mod 4, it's equivalent to a rotation of 90 degrees clockwise.

NB. All these rotations are rotations of the initial point (a,b) about the origin as the centre of rotation.