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Choice B is correct. The value of $f\left(144\right)$ is the value of $f\left(x\right)$ when $x=144$. It's given that the function $f$ is defined by $f\left(x\right)=4+\sqrt{x}$. Substituting $144$ for $x$ in this equation yields $f\left(144\right)=4+\sqrt{144}$. Since the positive square root of $144$ is $12$, it follows that this equation can be rewritten as $f\left(144\right)=4+12$, or $f\left(144\right)=16$. Therefore, the value of $f\left(144\right)$ is $16$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the value of $f\left(\mathrm{1,296}\right)$, not $f\left(144\right)$.

Choice D is incorrect. This is the value of $f\left(\mathrm{5,184}\right)$, not $f\left(144\right)$.