We present deep optical and near–infrared ( NIR ) UBVRIHKs imaging data for 24 blue compact galaxies ( BCGs ) . The individual exposure times are on average \sim 40 minutes in the optical ( B ) and \sim 90 minutes in the NIR , but on occasion up to \sim 5 hours for a single target and filter , observed on 2.5 ,~ { } 3.5 ,~ { } 8.2 –m telescopes . The sample contains luminous dwarf and intermediate–mass BCGs which are predominantly metal–poor , although a few have near–solar metallicities . We have analyzed isophotal and elliptical integration surface brightness and color profiles , extremely deep ( \mu _ { B } \lesssim 29 mag arcsec ^ { -2 } ) contour maps and RGB images for each galaxy in the sample , and provide a morphological classification where such is missing . We have measured the total galaxy colors , the colors of the underlying host galaxy , and the colors of the burst , and compare these to the predictions of new state–of–the–art spectral evolutionary models ( SEMs ) both with and without contribution by nebular emission . Separating the burst from the underlying host we find that regardless of the total luminosity the host galaxy has the properties of a low surface brightness ( LSB ) dwarf with M _ { B } \gtrsim - 18 . For a number of galaxies we discover a distinct LSB component dominant around and beyond the Holmberg radius . For the specific case of ESO400–43A & B we detect an optical bridge between the two companion galaxies at the \mu _ { V } \sim 28 th mag arcsec ^ { -2 } isophotal level . Synthetic disk tests are performed to verify that we can trace such faint components with negligible errors down to \mu _ { B } = 28 mag arcsec ^ { -2 } and \mu _ { K } = 23 mag arcsec ^ { -2 } . By examining the structural parameters ( central surface brightness \mu _ { 0 } and scale length h _ { r } ) derived from two radial ranges typically assumed to be dominated by the underlying host galaxy , we demonstrate the importance of sampling the host well away from the effects of the burst . We find that \mu _ { 0 } and h _ { r } of the BCGs host deviate from those of dwarf ellipticals ( dE ) and dwarf irregulars ( dI ) solely due to a strong burst contribution to the surface brightness profile almost down to the Holmberg radius . Structural parameters obtained from a fainter region , \mu _ { B } = 26 – 28 mag arcsec ^ { -2 } , are consistent with those of true LSB galaxies for the starbursting BCGs in our sample , and with dEs and dIs for the BCGs with less vigorous star formation .