Posterior reversible encephalopathy syndrome (PRES): This condition is strongly associated with hypertensive emergencies and can cause symptoms such as visual disturbances, headache, and seizures, although the patient's presentation of impaired horizontal gaze, multidirectional nystagmus, and impaired accommodation also points towards brainstem involvement which can be seen in posterior circulation infarcts. The high blood pressure and the presence of visual symptoms support this diagnosis.

Posterior circulation stroke: Given the patient's symptoms of giddiness, blurred vision, changes in color vision, and the examination findings of impaired horizontal gaze and multidirectional nystagmus, a stroke in the posterior circulation (including the brainstem) is a strong consideration. The patient's history of hypertension increases the risk of stroke.
Hypertensive encephalopathy: This condition occurs due to severely elevated blood pressure leading to cerebral edema and can present with symptoms such as headache, vision changes, and altered mental status. Although the patient does not have a headache or altered mental status, the high blood pressure and visual symptoms could still align with this diagnosis.

Subarachnoid hemorrhage: Although the patient does not have a thunderclap headache, which is classic for subarachnoid hemorrhage, the sudden onset of symptoms and severely elevated blood pressure could be indicative of a hemorrhagic stroke. Missing this diagnosis could be fatal.
Pituitary apoplexy: Given the patient's symptoms of visual disturbances and the presence of polydipsia and polyuria, which could indicate diabetes insipidus, pituitary apoplexy is a condition that should not be missed. It is a medical emergency that requires prompt treatment.
Cerebral vasculitis: This is an inflammation of the blood vessels in the brain and can present with a wide range of neurological symptoms, including visual disturbances and stroke-like symptoms. It is important to consider in patients with no clear risk factors for stroke or with atypical presentations.

Mitochondrial encephalomyopathy, lactic acidosis, and stroke-like episodes (MELAS): This is a rare mitochondrial disorder that can cause stroke-like episodes, often with a posterior circulation pattern. The patient's family history of inherited neurological conditions makes this a consideration, although it is rare.
Susac syndrome: A rare condition characterized by microangiopathic changes in the brain, retina, and cochlea, leading to a variety of symptoms including visual loss, hearing loss, and stroke-like episodes. It is more common in young women and could be considered given the patient's visual symptoms and the absence of other clear diagnoses.