Final answer:
The patient with a traumatic brain injury showing slow and confused responses but who can open his eyes and look at the examiner is most likely demonstrating a state called c) obtundation.
Step-by-step explanation:
A patient admitted to the hospital post-traumatic brain injury who can open his eyes, look at the examiner, but has slow and confused responses, is demonstrating signs of obtundation. Obtundation is characterized by a reduced level of consciousness with diminished responsiveness to the environment, and this level of arousal signals a moderate diminution in alertness, where the patient has a lessened interest in the environment, slowed responses to stimulation, and tends to be sleepy but can be a-r-o-u-s-e-d with moderate stimuli. According to the provided descriptions and options, the correct level of arousal for this patient would be c. Obtundation.