1. The problem asks to identify the locus of the point $z = x + iy$ in the complex plane defined by the equation $$|z-3| + |z+3i| = \text{constant}.$$\n\n2. Write $z = x + iy$, where $x,y \in \mathbb{R}$. Then $|z-3| = \sqrt{(x-3)^2 + y^2}$ is the distance from the point $(x,y)$ to $(3,0)$, and $|z+3i| = \sqrt{x^2 + (y+3)^2}$ is the distance from $(x,y)$ to $(0,-3)$.\n\n3. The sum of distances $|z-3| + |z+3i|$ from two fixed points $(3,0)$ and $(0,-3)$ to a point $(x,y)$ is constant. This is exactly the definition of an ellipse with foci at these two points.\n\nFinal Answer: The locus $|z-3|+|z+3i|=\text{constant}$ describes an ellipse with foci at $(3,0)$ and $(0,-3)$ in the complex plane.