Ah, it's a textured surface. And plastic. So you've got *both* problems - ink dye diffusing into the plastic, and dye sitting in the bottom of low points of the surface texture.
A similar problem I encounter is with that textured greenish vinyl Hewlett Packard laminated onto the aluminum panels of their instruments. Then people write on it with felt pens, damn them.
What I found works well, is a fine brass wire brush, and alcohol (methylated spirits) and/or turps. (Best mix depends on the type of marker pen.)
Literally pour some solvent on the surface then press hard with the brush while scrubbing in a drifting circular motion, so every groove of the texture gets bristles run along it in every direction - a lot.
The bristles get the ink out of texture grooves. It doesn't take too long to completely remove the marker.
But that might ruin the texture on that plastic case you have. If the dye has also diffused into the plastic, you may not be able to remove it AND keep the texture.
Edit: Oops, I see as someone else mentioned, it's actually painted metal. The ink dye probably wouldn't diffuse much into enamel, so maybe the wire brush method would work. Will roughen the paint surface, so then use a polish to get the shine back.
Or heck, you could just paint-strip to the metal then respray it?