the translation map z + a
The translation map is the simplest motion of the plane: f(z) = z + a, where a is a fixed complex number. It picks up every point and slides it by the same amount and in the same direction — exactly the vector a. Nothing rotates, nothing stretches, nothing flips; the whole plane just glides.
Concretely, if a = 2 - i then f shifts every point two units to the right and one unit down. Because z + a is the rule for adding a fixed vector, a shape and its image are congruent: same size, same orientation, same angles, merely relocated. A circle of radius 3 centred at the origin maps to a circle of radius 3 centred at a; a square maps to an identical square that has slid over. This is why translations are called rigid motions.
Translations are humble but indispensable. They are the building blocks you compose with rotations, scalings, and inversions to make every Mobius transformation, and they are the standard first step in normalizing a problem — slide your region so its centre sits at the origin, do the hard work there, then slide back. Recognizing 'this is just a translation' instantly tells you the geometry is unchanged.
Under f(z) = z + (3 + 0i) the point i maps to 3 + i, and the unit disk |z| <= 1 maps to the disk |w - 3| <= 1 — same disk, slid three units to the right.
A translation relocates without distorting; size, shape and orientation are all preserved.
A translation has no fixed point unless a = 0 (in which case it is the do-nothing identity map). This is in contrast to rotations and scalings, which always pin the origin in place.