I agree with the widespread consensus. Don't tell her. I have a couple of reasons that may differ a bit from other answers.
I'll give you my two reasons first, and then elaborate on both.
#1: A lot of other people (and including my former self) have ideas/beliefs about dreams and what their role is, and may be prone to judge dreams as having more weight than what I currently believe they should. Therefore, don't unnecessarily open yourself up for judgment for something that you have no control over, especially when you don't know what the other person's attitudes are about dreams.
#2: You say, Alice "has said on a few occasions how she's scared I'll leave her." Your relationship is not on solid ground. There is a shaky aspect to this. I think some relationships could weather such news just fine. Your relationship isn't there yet. Maybe tell Alice another day. Not now.
There. Those are my two reasons in a nutshell. First, I will briefly elaborate on my second point.
I've sometimes blabbed, only to find out soon after that such sharing resulted in causing unnecessary trouble and no significant benefit. Sometimes I've managed to just keep my mouth shut on some things, expecting that to be just temporary, only to soon find out that if I hadn't kept my mouth shut, the conversation would have turned out quite a bit worse. Do not conclude that you are lacking honesty just because you don't share information. Sometimes there may be some truth to the idea that you aren't being fully honest if you aren't being fully revealing, but that isn't always true. To quote Tom's answer, a "desire to be transparent is laudable, but in this case you want to unburden yourself, at her cost." Sometimes sharing information is more harmful than helpful. I'm thinking that would be the case if you tried sharing this detail now. Just learn to "bite your tongue" on this, at least until timing is more right (possibly years later). In a few weeks, you may look back and be glad you didn't blab. Or, if you blab, in a few minutes or semi-seconds you may find yourself quickly regretting such an irreversible action.
Are Dreams Legit?
And now, my lengthy justification to my first point... I will explain why I believe that dreams shouldn't have as much weight as what I once thought, and as many people currently think.
A lot of people don't understand dreams. But over my years, I've learned some things that I didn't know after my first couple of decades dreaming.
I remember being about 16 and having a dream that I remembered extremely vividly. I wrote it down, multiple pages. This dream had an erotically tantalizing element to it, involving a person that I never knew existed. Well, I must have subconsciously been exposed to this person's name somehow, otherwise how did I manage to have this person's name in my dream? When I got to school I checked and found out this person did exist. However, I obviously had no conscious feelings about this person whom I didn't recognize at all.
When I was nine or ten years of age, I committed suicide. Knife to the heart. Literally. All this, in a dream, of course. Once I was dead on the kitchen floor, I was surprised that I still felt pain from being stabbed by a knife. (I previously didn't think I would feel anything when dead.) Unfortunately, since I was dead, I couldn't pull the knife out of my heart to stop the pain. Since I was dead, I couldn't yell for help to my parents who were watching TV one room over, still completely unaware of what I did.
Any feelings of curiosity that my childhood self had about what it was like to die were put into check by that particular incident.
I remember for a few weeks I had dreams which were so realistic that, in the middle of my daytime hours, I recalled memories of recent events and only after thinking hard about them did I realize they were dreams. It started to concern me if my memory was unreliable, but fortunately that phase passed.
Other dreams had content notably different in life. I've done other things, and had other experiences, causing extreme pleasure (flying through the air) and notable turmoil. More than one dream has involved death... I remember when my dad died in a dream once, I cried myself to awakeness.
The dream-making component of my brain doesn't seem to like me analyzing things. Often when I've wanted to analyze some specific topic mid-dream, the dream conjures up some sort of compelling other topic for me to focus on instead.
However, more recently in my life, there's been several times now when the logic part of my brain won this war over attention, as I would refuse to be distracted at all or for more than a couple of seconds before I insist on focusing on one topic of interest that I really want to get figured out. What's happened is that whatever part of my brain was creating the dream responded by trying several times to introduce some new distraction, but eventually gave up, and my dream terminated. The next thing I knew, I opened my eyes and checked my clock as usual, and then realized that my scenery had just drastically changed as I just awoke from a dream. (I could often remember some of the recent details well enough for me to try completing my analysis of the recent situation without newly invented distractions trying to deter me.)
My current beliefs are that dreams are part of a sleep cycle that performs maintenance/repair of the brain. Sometimes my brain has come up with ludicrous scenarios of things that I would never do in real life. I think the brain is trying to stir up some specific emotion. As for why my brain does this, I don't know, because I don't fully understand dreaming or the biology of the human brain. Just as I don't understand why the brain is rather prone to forgetting dream content in a matter of minutes (or sometimes hours). But if the sleep process, including the dream process, is helpful for my health, and specifically some crucial life abilities like memory of things experienced while awake, then I am all in favor of these positive processes (despite not fully understanding them).
Being able to just "logic" myself to awakeness is something that only happened after probably 30+ years of never succeeding at focusing on a task that my dream-maker wanted to distract me from. I figure that I was probably just lightly sleeping at the time. During deeper sleep, the dream component of my brain probably has more free reign to succeed in creating all sorts of tremendous fantasies, both pleasant and terrible, without succumbing to interference by the annoying logic or moral decision-making components of my brain that would affect my decisions during my waking hours.
I know this analysis has been somewhat long and a bit off-topic from the questin's specifics, but I share this because I never logic'ed myself awake until later in life (30+ years), and never heard of this from anyone else before. When I was 20, I often believed that dreams may reflect unconscious desires. However, I now believe my dream making brain chemistry is just inventing whatever craziness it can in order to manipulate my emotions. Since this is all a part of natural sleep, I figure that somehow this ability of my dreams is probably somehow good for my health.
In dreams, I've dwelled on immoral lines of thinking, made immoral decisions, and performed immoral actions that I know I would never allow myself to accept in an awakened state. I've determined that I should not feel one bit guilty over dreams' content because such content can be entirely outside of any decision-making control that I have.
And that's why my current beliefs indicate that dream's contents shouldn't be judged. (But, to recap, since other people have other beliefs, don't go around sharing questionable content to people you closely know until you know what their beliefs are, and how they are likely to respond.)