Convert to Bibi-binary

Python 2, 58 bytes

f=lambda n:(n>15and f(n/16)or"")+"HBKD"[n/4%4]+"OAEI"[n%4]

A recursive solution. Try it on Ideone.

05AB1E, 20 18 16 bytes

hv…ÂkdžM¨ÁâyHèJ

Explanation

h                     # convert input to hex
 v                    # for each
  …Âkd               # string of the possible first Bibi-binary letters
       žM¨Á           # string of the possible second Bibi-binary letters
           â          # cartesian product to produce list of Bibi-binary pairs
            yH        # convert hex char to base 10
              è       # use this to index into the list
               J      # join

Try it online!

Saved 2 bytes thanks to Adnan

Python 2, 81 76 bytes

lambda n:''.join('HBKD'[int(x,16)/4]+'OAEI'[int(x,16)%4]for x in hex(n)[2:])

Chooses the bibi-digit to represent each hex digit based on the patterns in the bibi-digits.