There are a few ways you could do this.
Like @Vince suggested, you could draw a rectangle and use the Erase tool if your license level allows it.
Or, draw a rectangle of the same extent and perform a Union analysis and select the polygon in the middle (including the little islands).
Or, you could draw a rectangle of the same extent, then select your complex polygon. With your original polygon selected, select the Clip tool in the Editor toolbar:
Choose "Discard the area that intersects."
Then select the original polygon and delete it:
Another way you could do it would be to edit the vertices of your polygon and delete the four outer corners. This will give you the inverse polygon. Note, this might not always work, but it worked for this example.
Edit the vertices and select the four corners:
Delete the four vertices by right clicking on one of the selected ones:
Note, the result is a multipart polygon: