Clustering is a fundamental problem in unsupervised machine learning, and fair variants of it have recently received significant attention due to its societal implications. In this work we introduce a novel definition of individual fairness for clustering problems. Specifically, in our model, each point $j$ has a set of other points $\mathcal{S}_j$ that it perceives as similar to itself, and it feels that it is fairly treated if the quality of service it receives in the solution is $α$-close (in a multiplicative sense, for a given $α\geq 1$) to that of the points in $\mathcal{S}_j$. We begin our study by answering questions regarding the structure of the problem, namely for what values of $α$ the problem is well-defined, and what the behavior of the \emph{Price of Fairness (PoF)} for it is. For the well-defined region of $α$, we provide efficient and easily-implementable approximation algorithms for the $k$-center objective, which in certain cases enjoy bounded-PoF guarantees. We finally complement our analysis by an extensive suite of experiments that validates the effectiveness of our theoretical results.