**Core Concept**
One and a half syndrome is a rare neurological disorder characterized by a specific pattern of brainstem damage, typically resulting from a stroke or tumor. The name "one and a half" refers to the preservation of one internuclear oculomotor pathway, while the other is severely impaired, along with the loss of ipsilateral conjugate horizontal gaze.

**Why the Correct Answer is Right**
In one and a half syndrome, the medial longitudinal fasciculus (MLF) is damaged, causing an internuclear ophthalmoplegia (INO) on the ipsilateral side. However, the abducens nucleus and its connections to the lateral rectus muscle are spared, allowing for normal lateral gaze on the affected side. This is due to the fact that the MLF is responsible for coordinating conjugate horizontal eye movements between the two eyes, but the abducens nucleus controls the lateral rectus muscle directly.

**Why Each Wrong Option is Incorrect**
* **Option A:** Conjugate vertical gaze is typically spared in one and a half syndrome, as the superior and inferior colliculi and their connections are not affected. However, this option does not directly address the specific movement classically spared.
* **Option B:** Ipsilateral conjugate horizontal gaze is severely impaired in one and a half syndrome due to the damage to the MLF and the resulting INO.
* **Option D:** The option of sparing conjugate vertical gaze is incorrect because, although it's true, it doesn't address the movement classically spared, which is lateral gaze.

**Clinical Pearl / High-Yield Fact**
Remember the "ONE AND A HALF" acronym to help identify this condition: O (oculomotor nerve palsy), N (no adduction on the affected side), E (eyes deviate outward on the affected side), A (abducens nucleus spared), N (no nystagmus), D (dysconjugate eye movements), and H (horizontal gaze impaired).

**Correct Answer: C. Lateral gaze.**