Answer by emcconville for ImageMagick: convert to keep same name for converted image

Use the -set and formatting options.

convert *.psd -set filename:base "%[basename]""%[filename:base].png"

See "Long Form Attribute Percent Escapes" and "Filename Percent Escapes" docs.

Update

The mogrify utility that ships with imagemagick can also be used.

mogrify -format png *.psd

Note: Be careful with mogrify as the docs state...

This tool is similiar to convert except that the original image file is overwritten (unless you change the file suffix with the -format option) with any changes you request.