Canon 1D C has a Canon EF lens mount and currently there are 257 native lenses available for this mount.
The latest released lenses that are compatible with the 1D C are the Carl Zeiss Makro-Planar T2/50, the Canon 50mm f2.5 Macro and the Canon MP-E 65mm f2.5 1-5x Macro.
Canon 1D C doesn't have a sensor based image stabilization system but
77 of these lenses already comes with optical image stabilization.
Since Canon 1D C has a Full frame sensor, it has a focal length multiplier of 1x so lenses will provide their original focal length on 1D C.
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Canon 1D C Lenses with Image Stabilization
Canon 1D C Lenses with Weather Sealings