This point of confusion is discussed in Brhadaranyaka Upanishad. In the conversation between Yajnavalkya and his wife Maitreyi (2.4.12-14):
स यथा सैन्धवखिल्य उदके प्रास्त उदकमेवानुविलीयेत न हास्योद्ग्रहणायेव स्याद्यतो यतस्त्वाददीत लवणमेवैवं वा अर इदं महद्भूतमनन्तमपारं विज्ञानघन एवैतेभ्यो भूतेभ्यः समुत्थाय तान्येवानुविनश्यति न प्रेत्य संज्ञास्तीत्यरे ब्रवीमीति होवाच याज्ञवल्क्यः । सा होवाच मैत्रेय्यत्रैव मा भगवानमूमुहन्न प्रेत्य संज्ञास्तीति स होवाच याज्ञवल्क्यो न वा अरेऽहं मोहं ब्रवीम्यलं वा अर इदं विज्ञानाय । यत्र हि द्वैतमिव भवति तदितर इतरं जिघ्रति तदितर इतरं पश्यति तदितर इतरं शृणोति तदितर इतरमभिवदति तदितर इतरं मनुते तदितर इतरं विजानाति यत्र वा अस्य सर्वमात्मैवाभूत्तत्केन कं जिघ्रेत्तत्केन कं पश्येत्तत्केन कं शृणुयात्तत्केन कमभिवदेत्तत्केन कं मन्वीत तत्केन कं विजानीयात् येनेदं सर्वं विजानाति तं केन विजानीयाद्विज्ञातारमरे केन विजानीयात् ।
Just as a lump of salt when put into water becomes dissolved and cannot be taken out separately, but the water tastes salty throughout. Similarly this Great Existence, which is infinite and unbounded is a lump of consciousness. It arises from these creatures and gets dissolved when the creatures get dissolved (i.e. die). There is no consciousness after death.
Here Maitreyi said that she is confused by his statement that there is no consciousness after death.
To which, Yajnavalkya replies that he did not say anything confusing. This Brahman is indeed well-capable of cognition.
Where there is duality, there one smells, sees, hears, speaks with, thinks of, and knows another.
Where everything has become the Brahman or Atman, then whom and by what means should one smell, see, hear, speak with, think of, and know?
So the answer is that absolute consciousness/awareness/knowledge is the ultimate nature of Brahman. But in the "turiya" state, there is nothing but Brahman, so there is no second separate entity for it to know.
However, when the Brahman manifests as a mode in individuals, it can be differentiated by the contrast between the three states of consciousness, and hence recognized and realized.
This same paradox of the ultimate state of Brahman in-and-of-itself (i.e. "turiya") is also expressed in the Rig Veda's Nasadiya Suktam (RV 10.129.7):
यो अस्याध्यक्षः परमे व्योमन्त्सो अङ्ग वेद यदि वा न वेद
The Observer of all this, in the ultimate existence, even He may or may not know
This apparent (but not real) ambiguity is because we are describing both the dual and non-dual experience from the dual frame of reference. In the non-dual frame, there is nothing else to know, so Brahman does not know. But Brahman is knowledge itself.
Here's another perspective on this dichotomy: