The correct answer is option 'c) The Mahayana school'.

Explanation:
The Mahayana school of Buddhism serves as a bridge or halfway house between the old Buddhism and Hinduism. Here's why:

1. Emergence of Mahayana Buddhism:
- After the death of Gautama Buddha, Buddhism began to evolve and diversify into various schools.
- The Mahayana school emerged around the 1st century CE, during the period of the Kushanas.
- Mahayana means "Great Vehicle" and it aimed to make Buddhism accessible to a wider audience.

2. Influence of Hinduism on Mahayana Buddhism:
- Mahayana Buddhism absorbed several elements from Hinduism, which helped establish a connection between the two.
- Hindu deities such as Brahma, Vishnu, and Shiva were incorporated into the Mahayana pantheon of bodhisattvas.
- The concept of the divine Buddha, known as the Adi Buddha or the Primordial Buddha, was introduced in Mahayana Buddhism, resembling the Hindu notion of an eternal, transcendent deity.

3. Doctrine of Sunyata:
- Mahayana Buddhism introduced the concept of Sunyata, often translated as "emptiness" or "voidness".
- Sunyata emphasized the idea that all phenomena lack inherent existence or substantiality.
- This concept resonates with the Hindu concept of Maya, which suggests that the world is illusory and lacks ultimate reality.

4. Bodhisattva Ideal:
- Mahayana Buddhism places great emphasis on the Bodhisattva ideal, which involves working for the liberation of all sentient beings.
- This ideal aligns with the Hindu concept of a compassionate deity who assists others in their spiritual journey.

5. Scriptures and Practices:
- Mahayana Buddhism developed its own set of scriptures, known as the Mahayana Sutras, which included influential texts like the Lotus Sutra and the Heart Sutra.
- These texts often incorporated elements of Hindu philosophy and cosmology.
- Mahayana practices, such as chanting and devotion, bear similarities to Hindu devotional practices.

In conclusion, the Mahayana school of Buddhism forms a bridge between the old Buddhism and Hinduism due to its incorporation of Hindu deities, the introduction of the concept of Sunyata, the emphasis on the Bodhisattva ideal, and the inclusion of Hindu-influenced scriptures and practices.