**Core Concept:**
The question involves dysphagia, a clinical symptom characterized by difficulty in swallowing, and the bird-beak appearance on barium swallow, which is a specific feature in radiological examination. The question is asking about the appropriate diagnostic investigation for this clinical presentation.

**Why the Correct Answer is Right:**
The correct answer, **Option C: Endoscopy**, is the right choice due to its direct visualization of the oesophagus and the ability to identify various pathologies causing dysphagia. An endoscopy allows the physician to assess the mucosal lining of the oesophagus, detect lesions, inflammation, or strictures that may be responsible for the bird-beak appearance seen on a barium swallow.

**Why Each Wrong Option is Incorrect:**

A. **Option A: Barium swallow** is a crucial investigation for diagnosing swallowing disorders but does not have the capability to directly visualize the oesophagus and its underlying pathologies. It only gives information about the oesophagus's movement and the substance's passage through it.

B. **Option B: Computed Tomography (CT) Scan** is a valuable diagnostic tool for assessing various organs and structures but is less specific for oesophageal diseases compared to endoscopy. CT scans are primarily used for detecting lung, mediastinal, and chest abnormalities.

D. **Option D: X-ray** is a general term for imaging techniques that produce images of internal structures. However, it is not specific to the oesophagus and cannot directly visualize the oesophagus or assess for lesions causing the bird-beak appearance in a barium swallow.

**Clinical Pearl:**
In cases of progressive dysphagia and bird-beak appearance on barium swallow, endoscopy is the preferred diagnostic modality to directly visualize the oesophagus and identify the underlying pathology. This helps in accurate diagnosis, treatment planning, and appropriate referral for further management.