Can't even imagine the side effects of constantly growing non-self deleting cells would be however.
Oh, we already know that.
It's called 'cancer'. Hell, a lot (but not all) cancers *are* biologically immortal.
Assuming it locked my biological age to roughly where I was when I became immortal, then hell yes I'd take biological immortality in a heartbeat. If I were to continue becoming decrepit then I would have to think on it for a while.
Aging (*not* physical maturation, mind you, two separate processes) and biological immortality are pretty much mutually exclusive; aging is not a simple function of time, something that happens at a rate of a day per day of life, it's an accumulation of damage.
Keep aging and, well, you *will* die after a certain threshold. Any treatment for any age-related disease is just a stopgap. Cancer didn't get you? Well, let's try neurodegenerative disease. Or heart disease. Or infection, since your immune system is shot. Or trauma, since your bones aren't what they used to be.
So, at very least, biological immortality would require stopping the damage from accumulating, keeping you at the same age. Optimistically, we could reverse the damage already done, so you'd be locked to being somewhere in the range of twentysomething to thirty, mature but un-aged.
If we somehow achieved 100% reversal, you'd actually be *younger* purely aging-wise, than, say, a 6 y.o. currently is. Realistically, we would have some very odd-looking cases depending on what we managed to fix up, like someone looking like a perfectly normal adult except for liver spots and yellowed eyes.