Wherre are the adjacent characters in the title? [3, 4]!

Retina, 33 29 23 bytes

Saved 6 bytes thanks to Martin Ender

T`L`:`(.)\1+
:
$.`¶
T`L

Outputs a linefeed-separated list of indices.

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Explanation

T`L`:`(.)\1+

Transliterate runs of the same character into colons, to mark positions where there are duplicate characters.

:
$.`¶

Then replace each colon with the length of the text before it, followed by a linefeed.

T`L

Finally, delete any remaining letters.

Jelly, 7 bytes

JṁŒgḊÐf

1-based; returns a list of lists of the runs of indexes as allowed by the OP.

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How?

JṁŒgḊÐf - Main link: char-list s       e.g. 'DCCABBBACCCD' (which is a python interpreted input of ['D','C','C','A','B','B','B','A','C','C','C','D'])
J       - range(length(s))                  [1,2,3,4,5,6,7,8,9,10,11,12]
  Œg    - group-runs(s)                     [['D'],['C','C'],['A'],['B','B','B'],['A'],['C','C','C'],['D']]
 ṁ      - mould left like right             [[1],[2,3],[4],[5,6,7],[8],[9,10,11],[12]]
     Ðf - filter keep items that would be truthy (empty is not truthy) after applying:
    Ḋ   -     dequeue (x[1:])               [    [2,3],    [5,6,7],    [9,10,11]     ]        

Brain-Flak, 57 46 bytes

{({}[({})]<(([]<>)[()])>){(<{}{}{}>)}{}<>}<>

Includes +2 for -ar

Uses 0-based indexing.

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# While true
{

  # Subtract the second value on the stack from the first
  ({}[({})]

  # Push the height of this stack (the main stack) on the other stack
  <(([]<>)

  # Push the height of the main stack - 1
  [()])>

  # Push the difference that we calculated a second ago
  )

  # If they weren't the same character
  {

    # Pop the difference and the two stack heights
    (<{}{}{}>)

  # End if
  }

  # Pop the difference (or the 0 to get out of the if)
  {}

# Switch back to the main stack and end while
<>}

# Switch to the stack with the indexes and implicitly print
<>