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+ib, where i is the square root of -1. So, multiplying a+ib by -1, we get -a-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+ib by i, we get the complex number -b+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 i²=-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.