Using Total GIS Converter it should be possible to convert ArcExplorer project files to standard BMP files. However, it should be noted that BMP files are pretty large compared to let's say PNG or JPEG.
Convert ArcExplorer project file to Standard Windows bitmap image.
We found one software available in the database for this conversion.
Microsoft Windows software - convert aep to bmp on Windows
A converter that is able to transform various GIS file formats
Convert from aep file format
The table with program actions contains information about what each program is capable of doing with their files. This may be a good pointer for further searches for certain file conversions and help you find the converter you are looking for if our search result is insufficient for you. However, it is far from perfect and may sometimes show results which are not really usable because of the nature of the software handles the files. So a conversion that appears as possible, just because of matched actions, may in fact not be possible.